Bender groups as standard subgroups
Authors: Griess Robert L. , Mason David R. , Seitz Gary M. 
Published: July 2010

A subgroup X of a finite group G is called

∗

^ \ast

-standard if

X
~

=
X

/

O
(
X
)

\tilde X = X/O(X)

is quasisimple,

Y
=

C
G

(
X
)

Y = {C_G}(X)

is tightly embedded in G and

N
G

(
X
)
=

N
G

(
Y
)

{N_G}(X) = {N_G}(Y)

. This generalizes the notion of standard subgroups. Theorem. Let G be a finite group with

O
(
G
)
=
1

O(G) = 1

. Suppose X is

∗

^ \ast

-standard in G and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X overTilde slash upper Z left-parenthesis upper X overTilde right-parenthesis approximately-equals upper L 2 left-parenthesis 2 Superscript n Baseline right-parenthesis comma upper U 3 left-parenthesis 2 Superscript n Baseline right-parenthesis">

X
~

/

Z
(

X
~

)
≅

L
2

(

2
n

)
,

U
3

(

2
n

)

\tilde X/Z(\tilde X) \cong {L_2}({2^n}),{U_3}({2^n})

or

Sz

(

2
n

)

{\text {Sz}}({2^n})

. Assume

X
⋪
G

X \ntriangleleft G

. Then

O
(
X
)
=
1

O(X) = 1

and one of the following holds:

(

i

)

E
(
G
)
≅
X
×
X

({\text {i}})\;E(G) \cong X \times X

.

(

ii

)

X
≅

L
2

(

2
n

)

({\text {ii}})\;X \cong {L_2}({2^n})

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis upper G right-parenthesis approximately-equals upper L 2 left-parenthesis 2 Superscript 2 n Baseline right-parenthesis comma upper U 3 left-parenthesis 2 Superscript n Baseline right-parenthesis o r upper L 3 left-parenthesis 2 Superscript n Baseline right-parenthesis">

E
(
G
)
≅

L
2

(

2

2
n

)
,

U
3

(

2
n

)

o
r

L
3

(

2
n

)

E(G) \cong {L_2}({2^{2n}}),{U_3}({2^n})\;or\;{L_3}({2^n})

.

(

iii

)

X
≅

U
3

(

2
n

)

({\text {iii}})\;X \cong {U_3}({2^n})

and

E
(
G
)
≅

L
3

(

2

2
n

)

E(G) \cong {L_3}({2^{2n}})

.

(

iv

)

X
≅

Sz

(

2
n

)

({\text {iv}})\;X \cong {\text {Sz}}({2^n})

and

E
(
G
)
≅

Sp

(
4
,

2
n

)

E(G) \cong {\text {Sp}}(4,{2^n})

.

(

v

)

X
≅

L
2

(
4
)

({\text {v}})\;X \cong {L_2}(4)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis upper G right-parenthesis approximately-equals upper M 12 comma upper A 9 comma upper J 1 comma upper J 2 comma upper A 7 comma upper L 2 left-parenthesis 25 right-parenthesis comma upper L 3 left-parenthesis 5 right-parenthesis o r upper U 3 left-parenthesis 5 right-parenthesis">

E
(
G
)
≅

M

12

,

A
9

,

J
1

,

J
2

,

A
7

,

L
2

(
25
)
,

L
3

(
5
)

o
r

U
3

(
5
)

E(G) \cong {M_{12}},{A_9},{J_1},{J_2},{A_7},{L_2}(25),{L_3}(5)\;or\;{U_3}(5)

.

(

vi

)

X
≅

Sz

(
8
)

({\text {vi}})\;X \cong {\text {Sz}}(8)

and

E
(
G
)
≅

Ru

E(G) \cong {\text {Ru}}

(the Rudvalis group).

(

vii

)

X
≅

L
2

(
8
)

({\text {vii}})\;X \cong {L_2}(8)

and

E
(
G
)
≅

G
2

(
3
)

E(G) \cong {G_2}(3)

.

(

viii

)

X
≅

SL

(
2
,
5
)

({\text {viii}})\;X \cong {\text {SL}}(2,5)

and G has sectional 2-rank at most 4. In particular, if G is simple,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G approximately-equals upper M 12 comma upper A 9 comma upper J 1 comma upper J 2 comma Ru comma upper U 3 left-parenthesis 5 right-parenthesis comma upper L 3 left-parenthesis 5 right-parenthesis comma upper G 2 left-parenthesis 5 right-parenthesis comma o r cubed upper D 4 left-parenthesis 5 right-parenthesis">

G
≅

M

12

,

A
9

,

J
1

,

J
2

,

Ru

,

U
3

(
5
)
,

L
3

(
5
)
,

G
2

(
5
)
,
o
r

3

D
4

(
5
)

G \cong {M_{12}},{A_9},{J_1},{J_2},{\text {Ru}},{U_3}(5),{L_3}(5),{G_2}(5), or\;{^3}{D_4}(5)

.