<PAGE>   1

                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                    FORM 10-Q

(Mark One)


[X]             QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF
                       THE SECURITIES EXCHANGE ACT OF 1934

                FOR THE QUARTERLY PERIOD ENDED SEPTEMBER 30, 1998

                                       OR
[ ]             TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF
                       THE SECURITIES EXCHANGE ACT OF 1934

            FOR THE TRANSITION PERIOD FROM __________ TO ___________

                        COMMISSION FILE NUMBER: 33-45417

                              THE BISYS GROUP, INC.

             (Exact name of registrant as specified in its charter)

<TABLE>
<S>                                                                <C>       
                         DELAWARE                                               13-3532663
(State or other jurisdiction of incorporation or organization)     (I.R.S. Employer Identification No.)
</TABLE>

                    150 CLOVE ROAD, LITTLE FALLS, NEW JERSEY
                                      07424

                    (Address of principal executive offices)
                                   (Zip Code)

                                  973-812-8600

              (Registrant's telephone number, including area code)


INDICATE BY CHECK MARK WHETHER THE REGISTRANT (1) HAS FILED ALL REPORTS REQUIRED
TO BE FILED BY SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE ACT OF 1934 DURING
THE PRECEDING 12 MONTHS (OR FOR SUCH SHORTER PERIOD THAT THE REGISTRANT WAS
REQUIRED TO FILE SUCH REPORT(S), AND (2) HAS BEEN SUBJECT TO SUCH FILING
REQUIREMENTS FOR THE PAST 90 DAYS.

YES   X    NO ___

INDICATE THE NUMBER OF SHARES OUTSTANDING OF EACH OF THE ISSUER'S CLASSES OF
COMMON STOCK, AS OF THE LATEST PRACTICABLE DATE:

<TABLE>
<CAPTION>
               Class                              Shares Outstanding at October 30, 1998 
<S>                                               <C>                                
Common Stock, par value $.02 per share                           26,631,882                         
</TABLE>

                        This document contains 13 pages.

<PAGE>   2
                              THE BISYS GROUP, INC.

                               INDEX TO FORM 10-Q




<TABLE>
<CAPTION>
PART I.  FINANCIAL INFORMATION                                                       Page

<S>                                                                                  <C>
    Item 1.  Financial Statements

             Condensed Consolidated Balance Sheet as of September 30, 1998
               and June 30, 1998                                                       3

             Condensed Consolidated Statement of Operations for the three months       4
               ended September 30, 1998 and 1997

             Condensed Consolidated Statement of Cash Flows for the three months       5
               ended September 30, 1998 and 1997

             Notes to Condensed Consolidated Financial Statements                      6


    Item 2.  Management's Discussion and Analysis of Results of Operations and         8
             Financial Condition



PART II.  OTHER INFORMATION

    Item 2.  Changes in Securities and Use of Proceeds                                11

    Item 6.  Exhibits and Reports on Form 8-K                                         11

SIGNATURES                                                                            12

EXHIBIT INDEX                                                                         13
</TABLE>


                                       2
<PAGE>   3
                                     PART I

                          ITEM 1. FINANCIAL STATEMENTS

                     THE BISYS GROUP, INC. AND SUBSIDIARIES
                      CONDENSED CONSOLIDATED BALANCE SHEET
                  (DOLLARS IN THOUSANDS, EXCEPT PER SHARE DATA)
                                   (UNAUDITED)

<TABLE>
<CAPTION>
                                                                         September 30,   June 30,
                                                                             1998          1998
                                                                          ---------      ---------
<S>                                                                      <C>             <C>      
ASSETS

Current assets:
  Cash and cash equivalents                                               $  39,291      $  93,403
  Accounts receivable, net                                                   80,263         73,693
  Deferred tax asset                                                          5,179          4,660
  Prepaid expenses and other                                                 14,268          8,484
                                                                          ---------      ---------
     Total current assets                                                   139,001        180,240
Property and equipment, net                                                  45,180         37,478
Intangible assets, net                                                      144,080        102,663
Other assets                                                                 21,227         13,720
                                                                          ---------      ---------
     Total assets                                                         $ 349,488      $ 334,101
                                                                          =========      =========

LIABILITIES AND STOCKHOLDERS' EQUITY
Current liabilities
  Current maturities of long-term debt                                    $     202      $     124
  Accounts payable                                                           15,602         11,626
  Short-term borrowings                                                      10,000           --
  Other current liabilities                                                  77,388         70,668
                                                                          ---------      ---------
     Total current liabilities                                              103,192         82,418
Long-term debt                                                                2,404          1,578
Deferred tax liability                                                        9,260         10,451
Other liabilities                                                             5,203          1,364
                                                                          ---------      ---------
     Total liabilities                                                      120,059         95,811
                                                                          ---------      ---------

Stockholders' equity:
  Common stock, $.02 par value, 80,000,000 shares authorized,
  26,670,388 shares issued                                                      533            533
  Additional paid-in capital                                                179,096        173,683
  Retained earnings                                                          51,415         66,229
  Less treasury stock at cost, 40,970 and 57,895 shares, respectively        (1,615)        (2,155)
                                                                          ---------      ---------
     Total stockholders' equity                                             229,429        238,290
                                                                          ---------      ---------
     Total liabilities and stockholders' equity                           $ 349,488      $ 334,101
                                                                          =========      =========
</TABLE>


The accompanying notes are an integral part of the condensed consolidated
financial statements.


                                       3
<PAGE>   4
                     THE BISYS GROUP, INC. AND SUBSIDIARIES
                 CONDENSED CONSOLIDATED STATEMENT OF OPERATIONS
                      (IN THOUSANDS, EXCEPT PER SHARE DATA)
                                   (UNAUDITED)


<TABLE>
<CAPTION>
                                                       Three Months Ended
                                                         September 30,
                                                   ------------------------
                                                      1998           1997
                                                   ---------      ---------
<S>                                                <C>            <C>      
Revenues                                           $ 101,924      $  91,462
                                                   ---------      ---------
Operating costs and expenses:
  Service and operating                               59,872         54,218
  General and administrative                          16,385         15,606
  Selling and conversion                               5,039          4,156
  Research and development                             3,181          2,871
  Amortization of intangible assets                    1,508            888
  Merger expenses and other charges                      400         11,998
  Acquired in-process research and development        19,000           --   
                                                   ---------      ---------
Operating earnings (loss)                             (3,461)         1,725
Interest income, net                                     797          1,025
                                                   ---------      ---------
Income (loss) before income taxes                     (2,664)         2,750
Income taxes                                           6,535          1,127
                                                   ---------      ---------
Net income (loss)                                  $  (9,199)     $   1,623
                                                   =========      =========

Basic earnings (loss) per share                    $   (0.35)     $    0.06
                                                   =========      =========

Diluted earnings (loss) per share                  $   (0.35)     $    0.06
                                                   =========      =========
</TABLE>



The accompanying notes are an integral part of the condensed consolidated
financial statements.


                                       4
<PAGE>   5
                     THE BISYS GROUP, INC. AND SUBSIDIARIES
                 CONDENSED CONSOLIDATED STATEMENT OF CASH FLOWS
                                 (IN THOUSANDS)
                                   (UNAUDITED)

<TABLE>
<CAPTION>
                                                                                     Three Months Ended
                                                                                       September 30,
                                                                                   ----------------------
                                                                                       1998          1997
                                                                                   --------      --------
<S>                                                                                <C>           <C>     
Cash flows from operating activities:
Net earnings                                                                       $ (9,199)     $  1,623
Adjustments to reconcile net earnings to net cash provided
by operating activities:
  Depreciation and amortization                                                       4,903         3,472
  Loss on disposition or write-down of property and equipment                          --           2,520
  Write-off of acquired in process research and development                          19,000          --
  Deferred income tax provision (benefit)                                              --            (540)
  Change in operating assets and liabilities, net of effects from acquisitions      (11,284)        6,429
                                                                                   --------      --------
Net cash provided by operating activities                                             3,420        13,504
                                                                                   --------      --------

Cash flows from investing activities:
  Acquisition of businesses                                                         (20,340)         --
  Net cash acquired in acquisitions                                                   4,554         1,490
  Capital expenditures                                                               (7,468)       (5,377)
  Proceeds from sales of investments                                                   --           1,203
  Purchase of investments                                                            (1,067)       (5,000)
  Other                                                                                 285            39
                                                                                   --------      --------
Net cash used in investing activities                                               (24,036)       (7,645)
                                                                                   --------      --------

Cash flows from financing activities:
  Proceeds from borrowings                                                           10,000          --
  Repayment of debt                                                                     (40)       (2,043)
  Proceeds from exercise of stock options                                             2,327         1,685
  Repurchases of common stock                                                       (45,783)         --
                                                                                   --------      --------
Net cash used in financing activities                                               (33,496)         (358)
                                                                                   --------      --------
Net increase (decrease) in cash and cash equivalents                                (54,112)        5,501

Cash and cash equivalents at beginning of period                                     93,403        79,951
                                                                                   --------      --------
Cash and cash equivalents at end of period                                         $ 39,291      $ 85,452
                                                                                   ========      ========
</TABLE>


The accompanying notes are an integral part of the condensed consolidated
financial statements.


                                       5
<PAGE>   6
                          THE BISYS GROUP, INC. AND SUBSIDIARIES
                   NOTES TO CONDENSED CONSOLIDATED FINANCIAL STATEMENTS
                                       (UNAUDITED)

1.   THE COMPANY

     The BISYS(R) Group, Inc. and subsidiaries (the "Company") is a leading
     national provider of outsourcing solutions to and through financial
     organizations.

     The condensed consolidated financial statements include the accounts of The
     BISYS Group, Inc. and its subsidiaries and have been prepared consistent
     with the accounting policies reflected in the 1998 Annual Report on Form
     10-K filed with the Securities and Exchange Commission and should be read
     in conjunction therewith. The condensed consolidated financial statements
     include all adjustments (consisting only of normal recurring adjustments)
     which are, in the opinion of management, necessary to present fairly this
     information.

2.   USE OF ESTIMATES

     The preparation of financial statements in conformity with generally
     accepted accounting principles requires management to make certain
     estimates and assumptions that affect the reported amounts of assets and
     liabilities and disclosure of contingent assets and liabilities at the date
     of the financial statements and the reported amounts of revenue and
     expenses during the reporting period. The most significant estimates are
     related to the allowance for doubtful accounts, intangible assets, merger
     expenses and other charges, income taxes and contingencies. Actual results
     could differ from these estimates in the near term.

3.   EARNINGS PER SHARE

     Basic and diluted EPS computations for the three months ended September 30,
     1998 and 1997 are as follows (in thousands, except per share amounts):

<TABLE>
<CAPTION>
                                      Three Months Ended
                                         September 30,
                                    ----------------------
                                      1998          1997
                                    --------      --------
<S>                                 <C>           <C>     
Basic EPS
-------------
Net income (loss)                   $ (9,199)     $  1,623
                                    ========      ========

Weighted average common shares
outstanding                           26,351        26,104
                                    ========      ========

Basic earnings (loss) per share     $  (0.35)     $   0.06
                                    ========      ========
</TABLE>


<TABLE>
<CAPTION>
                                         Three Months Ended
                                            September 30,
                                        ----------------------
                                          1998          1997
                                        --------      --------
<S>                                     <C>           <C>     
Diluted EPS
--------------
Net income (loss)                       $ (9,199)     $  1,623
                                        ========      ========

Weighted average common shares
outstanding                               26,351        26,104

Assumed conversion of common shares
issuable under stock option plans           --           1,331
                                        --------      --------

Weighted average common and common
equivalent shares outstanding             26,351        27,435
                                        ========      ========

Diluted earnings (loss) per share       $  (0.35)     $   0.06
                                        ========      ========
</TABLE>


                                       6
<PAGE>   7


     Options to purchase 4,552,920 shares of common stock at various prices
     ranging from $0.03 to $44.50 were outstanding at September 30, 1998, but
     were not included in the computation of diluted EPS because of the net loss
     for the period.

4.   BUSINESS COMBINATIONS

     On July 16, 1998, the Company completed its acquisition of Corelink
     Resources, Inc. (Corelink), which the Company has held a minority interest
     in since fiscal 1995. On August 10, 1998, the Company acquired Potomac
     Insurance Marketing Group, Inc. (Potomac). Both transactions have been
     accounted for by the purchase method of accounting and involved total cash
     consideration of $20.3 million. The operations of both Corelink and Potomac
     are included in the consolidated results of operations from the dates of
     acquisition. Pro forma information has not been presented due to lack of
     materiality.

     On September 16, 1998, the Company completed its acquisition of Greenway
     Corporation (Greenway) through the issuance of 968,202 shares of BISYS
     common stock held in treasury and issuance of 148,795 equivalent stock
     options for all of the outstanding shares and stock options of Greenway.
     The acquisition, valued at approximately $43.8 million, was treated as a
     purchase for accounting purposes, and, accordingly, the assets and
     liabilities were recorded based on their fair values at the date of the
     acquisition. Of the total purchase price, $15.6 million was allocated to
     goodwill, $7.4 million to other identifiable intangible assets, and $1.8
     million to net tangible assets. In addition, $19.0 million was allocated to
     acquired in-process research and development, which was charged to
     operations at the time of the acquisition. Greenway's operations are
     included in the consolidated results of operations from the date of the
     acquisition. Pro forma information has not been presented due to lack of
     materiality.


                                       7
<PAGE>   8
ITEM 2.             MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                 RESULTS OF OPERATIONS AND FINANCIAL CONDITION

The Company provides outsourcing solutions to and through financial
organizations which is reported as a single segment. The operating margins for
each business unit of the Company are not significantly different. The following
table presents the percentage of revenues represented by each item in the
Company's condensed consolidated statement of operations for the periods
indicated:

<TABLE>
<CAPTION>
                                                       Three Months Ended
                                                         September 30,
                                                      -------------------
                                                       1998         1997
                                                      ------       ------
<S>                                                   <C>          <C>   
Revenues                                              100.0%       100.0%
                                                      -----        ----- 

Operating costs and expenses:
  Service and operating                                58.7         59.3
  General and administrative                           16.1         17.1
  Selling and conversion                                4.9          4.5
  Research and development                              3.1          3.1
  Amortization of intangible assets                     1.5          1.0
  Merger expenses and other charges                     0.4         13.1
  Acquired in-process research and development         18.7          --   
                                                      -----        ----- 
Operating earnings (loss)                              (3.4)         1.9
Interest income, net                                    0.8          1.1
                                                      -----        ----- 
Income (loss) before income taxes                      (2.6)         3.0
Income taxes                                            6.4          1.2
                                                      -----        ----- 
Net income (loss)                                      (9.0)%        1.8%
                                                      =====        =====
</TABLE>


COMPARISON OF THE THREE MONTHS ENDED SEPTEMBER 30, 1998 WITH THE THREE MONTHS
ENDED SEPTEMBER 30, 1997.

      Revenues increased 11.4% from $91.5 million for the three months ended
      September 30, 1997 to $101.9 million for the three months ended September
      30, 1998. This growth was derived from sales to new clients, existing
      client growth, cross sales to existing clients and revenues from acquired
      businesses, partially offset by lost business.

      Service and operating expenses increased 10.4% from $54.2 million during
      the three months ended September 30, 1997 to $59.9 million for three
      months ended September 30, 1998, and decreased as a percentage of revenues
      from 59.3% to 58.7%. The dollar increase resulted from additional costs
      associated with greater revenues.

      General and administrative expenses increased 5.0% from $15.6 million
      during the three months ended September 30, 1997, to $16.4 million for the
      three months ended September 30, 1998, and decreased as a percentage of
      revenues from 17.1% to 16.1%. The dollar increase primarily resulted from
      additional costs associated with recent acquisitions. The decrease as a
      percentage of revenues resulted from further utilization of existing
      general and administrative support resources.

      During the three months ended September 30, 1998, the Company wrote off
      $19.0 million of acquired in-process research and development associated
      with the acquisition of Greenway and incurred $0.4 million of
      merger-related expenses.

      The income tax provision of $6.5 million for the three months ended
      September 30, 1998 increased from $1.1 million for the three months ended
      September 30, 1997 primarily due to higher taxable income. The provision
      represents an effective tax rate of 40% for the three months ended
      September 30, 1998, excluding the nonrecurring nondeductible charge of
      $19.0 million related to acquired in-process research and development,
      compared to an effective tax rate of 41% for the three months ended
      September 30, 1997.


                                       8
<PAGE>   9
LIQUIDITY AND CAPITAL RESOURCES

      At September 30, 1998, the Company had cash and cash equivalents of $39.3
      million and working capital of approximately $35.8 million. During the
      three months ended September 30, 1998, the Company borrowed $10.0 million
      against its revolving credit facility to support working capital
      requirements and fund certain acquisitions completed in the first fiscal
      quarter. The credit facility bears interest at LIBOR plus a margin not to
      exceed 1.25%, which rate aggregated 6.25% at September 30, 1998. At
      September 30, 1998, the Company had $0.6 million outstanding in the form
      of letters of credit. The weighted average interest rate on other
      outstanding long-term borrowings of $2.6 million at September 30, 1998 was
      7.82%.

      For the three months ended September 30, 1998, operating activities
      provided cash of $3.4 million. Investing activities used cash of $24.0
      million primarily for the acquisition of businesses of $20.3 million,
      capital expenditures of $7.5 million and investments of $1.1 million
      offset by cash acquired of $4.6 million. Financing activities used cash of
      $33.5 million primarily for the repurchase of common stock of $45.8
      million offset by borrowings of $10 million and $2.3 million of proceeds
      from the exercise of stock options.

      In accordance with the Company's previously announced stock buy-back
      program, the Company purchased approximately 1.1 million shares of its
      common stock for approximately $45.8 million in order to effect the
      acquisition of Greenway and for issuance of stock in connection with the
      exercise of stock options.

MERGER EXPENSES AND OTHER CHARGES

      At September 30, 1998, approximately $2.1 million of costs to integrate
      new operations arising from prior acquisitions and costs relating to the
      realignment of certain operations are included in accrued liabilities on
      the accompanying balance sheet. Approximately $1.4 million of such
      expenses were paid or charged against the related accruals during the
      three months ended September 30, 1998.

YEAR 2000

      The Company is addressing the Year 2000 issues associated with its
      existing computer systems and software applications utilizing both
      internal and external resources to identify and remediate these matters
      throughout the organization. The Company has completed its risk assessment
      and continues to remediate significant systems which are not currently
      Year 2000 ready. The Company anticipates that all of its internal mission
      critical information systems will be Year 2000 ready by December 31, 1998.
      In the event such systems are not Year 2000 ready by December 31, 1999, it
      could have a material adverse impact on the Company's business and results
      of operations.

      The Company uses third party provided software and systems for such tasks
      as account and information statement processing, fund accounting, and
      401(k) plan record keeping. If third parties upon which the Company
      depends are unable to address their Year 2000 issues in a timely manner,
      it could result in a material adverse financial risk to the Company. In
      order to assure that this does not occur, the Company is devoting
      resources necessary to develop appropriate business continuity plans.
      These contingency plans will include alternative systems and vendors,
      disaster recovery hot sites and manual processes. These contingency plans
      are expected to be completed prior to June 30, 1999.

      The Company's Year 2000 progress, the testing of remediated software, and
      contingency plans have been and will continue to be the subject of
      independent verification and validation by the Company's Internal Audit
      function. Internal Audit reports on Year 2000 are reviewed by senior
      management and the Company's Board of Directors.

      The Company believes it has developed an effective plan to address the
      Year 2000 issues and that, based on available information, its Year 2000
      transition will not have a material effect on its business, operations, or
      financial results. The Company anticipates expenditures for Year 2000
      testing and remediation in the range of $3.0 to $4.0 million in fiscal
      1999 and less than $1.5 million in the first half of fiscal 2000.

SAFE HARBOR STATEMENT UNDER THE PRIVATE SECURITIES LITIGATION REFORM ACT OF 1995

      Except for the historical information contained herein, the matters
      discussed in this quarterly report are forward-looking statements which
      involve risks and uncertainties, including but not limited to economic,


                                       9
<PAGE>   10
      competitive, governmental and technological factors affecting the
      Company's operations, markets, services and related products, prices, and
      other factors discussed in the Company's prior filings with the Securities
      and Exchange Commission. Although the Company believes that the
      assumptions underlying the forward-looking statements contained herein are
      reasonable, any of the assumptions could be inaccurate. Therefore, there
      can be no assurance that the forward-looking statements included in this
      quarterly report will prove to be accurate. In light of the significant
      uncertainties inherent in the forward-looking statements included herein,
      the inclusion of such information should not be regarded as a
      representation by the Company or any other person that the objectives and
      plans of the Company will be achieved.


                                       10
<PAGE>   11
                                     PART II

ITEM 2.     CHANGES IN SECURITIES AND USE OF PROCEEDS

            On September 16, 1998, the registrant issued an aggregate of 968,202
            shares of its common stock, $.02 par value ("Registrant Common
            Stock"), to the stockholders of Greenway Corporation, a Georgia
            corporation ("Greenway"), in connection with the acquisition of
            Greenway through the merger of a wholly-owned subsidiary of the
            registrant with and into Greenway (the "Merger"). Pursuant to the
            Agreement and Plan of Merger entered into with respect to the
            Merger, the 542,276 shares of Greenway common stock outstanding on
            the effective date of the Merger were converted into shares of
            Registrant Common Stock valued at $42.5238 per share. Said shares of
            Registrant Common Stock were not registered under the Securities Act
            of 1933, as amended (the "Securities Act"). There was no underwriter
            or placement agent.

            In connection with the issuance of shares of Registrant Common Stock
            to the stockholders of Greenway in the Merger, the registrant relied
            on an exemption from registration under Section 4(2) of the
            Securities Act, based, among other things, on certain
            representations and warranties of the investors, and information
            provided to the investors with respect to the registrant, Greenway
            and the Merger.

ITEM 6.     EXHIBITS AND REPORTS ON FORM 8-K

      (a)   EXHIBITS

            Exhibit 2 - Agreement and Plan of Merger, dated as of September 16,
            1998, as amended, among the Registrant and Greenway Corporation.
            (Reference is made to Exhibit 2.1 and 2.2 to the Current Report on
            Form 8-K, dated September 16, 1998, filed with the Securities and
            Exchange Commission, and incorporated herein by reference.)

      (b)   REPORTS ON FORM 8-K

            A current report on Form 8-K, dated as of September 16, 1998, was
            filed with the Securities and Exchange Commission on September 22,
            1998 (Items 2 and 7).


                                       11
<PAGE>   12
                                   SIGNATURES


Pursuant to the requirements of the Securities Exchange Act of 1934, the
Registrant has duly caused this report to be signed on its behalf by the
undersigned, thereunto duly authorized.


                                     THE BISYS GROUP, INC.



Date: November 13, 1998              By: /s/ Dennis R. Sheehan
      -----------------              -------------------------------------
                                             Dennis R. Sheehan
                                             Executive Vice President and
                                             Chief Financial Officer
                                             (Duly Authorized Officer)


                                       12
<PAGE>   13
                              THE BISYS GROUP, INC.
                                  EXHIBIT INDEX



EXHIBIT NO.                                                  PAGE

      (27)  Financial Data Schedule.........................(electronic only)


                                       13