characterization of ordered groups of rank one

characterization of ordered groups of rank one

For an

ordered group

, having rank (

http://planetmath.org/IsolatedSubgroup

) one is to an

Archimedean property

. In this entry, we use multiplicative notation for groups.

Lemma

An ordered group has rank one if and only if, for every two

elements

x

and

y

such that

x

<

y

<

1

, there exists an integer

n

>

1

such that

y

n

<

x

.

Proof

Suppose that the Archimedean property is satisfied and that

F

is an

isolated subgroup

of

G

. We shall show that if

F

contains any element other than the

identity

, then

F

=

G

. First note that there must exist an

x

∈

F

such that

x

<

1

. By

assumption

, there must exist an element

x

′

∈

F

such that

x

′

≠

1

. By

conclusion

1 of the basic

theorem

on ordered groups, either

x

′

<

1

, or

x

′

>

1

(since we assumed that the case

x

′

=

1

is excluded). If

x

′

<

1

, set

x

=

x

′

. If not, by conclusion 5, if

x

′

>

1

, then we will have

x

′

⁣

-

1

<

0

and therefore will set

x

=

x

′

⁣

-

1

when

x

>

1

.

Let

y

be any element of

G

. There are five possibilities:

1.

y

=

1

2.

x

=

y

3.

x

<

y

<

1

4.

y

<

x

<

1

5.

1

<

y

We shall show that in each of these cases,

y

∈

F

.

1.

Trivial

— 1 is an element of every group.

2.

Trivial —

x

is assumed to belong to

F

3.

Since

F

is an isolated subgroup,

y

∈

G

.

4.

By the Archimedean property,there exists an integer

n

such that

x

n

<

y

<

1

. Since

x

n

∈

F

and

F

is isolated (

http://planetmath.org/IsolatedSubgroup

), it follows that

y

∈

F

.

5.

1

<

y

By conclusion 5 of the basic theorem on ordered groups,

y

-

1

<

1

. By conclusion 1 of the same theorem, either

y

-

1

<

x

or

y

-

1

=

1

or

x

<

y

. In each of these three cases, it follows that

y

-

1

∈

F

from what we have already shown. Since

F

is a group,

y

-

1

∈

F

implies

y

∈

F

.

This shows that the only isolated subgroups of

G

are the two

trivial subgroups

(i.e. the group

{

1

}

and

G

itself), and hence

G

has rank one.

Next, suppose that

G

does not enjoy the Archimedean property. Then there must exist

x

∈

G

and

y

∈

G

such that

x

<

y

n

<

1

for all integers

n

>

0

. Define the sets

F

n

as

F

n

=

{

z

∈

G

∣

y

n

≦

z

≦

y

-

n

}

and define

F

=

⋃

n

=

1

∞

F

n

.

We shall show that

F

is a

subgroup

of

G

. First, note that, by a corollary of the basic theorem on ordered groups,

y

n

<

1

<

y

, so

1

∈

F

n

for all

n

, hence

1

∈

F

. Second, suppose that

z

∈

F

n

. Then

y

n

≦

z

≦

y

-

n

. By conclusion 5 of the basic theorem,

y

n

≦

z

implies

z

-

1

≦

y

-

n

and

z

≦

y

-

n

implies

y

n

≦

z

-

1

. Thus,

y

n

≦

z

-

1

≦

y

-

n

, so

z

-

1

∈

F

n

. Hence, if

z

∈

F

, then

z

-

1

∈

F

. Third, suppose that

z

∈

F

and

w

∈

F

. Then there must exist integers

m

and

n

such that

z

∈

F

n

and

w

∈

F

m

, so

y

n

≦

z

≦

y

-

n

and

y

m

≦

w

≦

y

-

m

.

Using conclusion 4 of the main theorem repeatedly, we conclude that

y

m

+

n

≦

z

⁢

w

≦

y

-

m

-

n

so

z

⁢

w

∈

F

m

+

n

. Hence, if

z

∈

F

and

w

∈

F

, then

z

⁢

w

∈

F

. this the proof that

F

is a subgroup of

G

.

Not only is

F

a subgroup of

G

, it is an isolated subgroup. Suppose that

f

∈

F

and

g

∈

G

and

f

≦

g

≦

1

. Since

f

∈

F

, there must exist an

n

such that

f

∈

F

n

, hence

y

n

≦

f

. By conclusion 2 of the basic theorem on ordered groups,

y

n

≦

f

and

f

≦

g

imply

y

n

≦

g

. Combining this with the facts that

g

≦

1

and

1

≦

y

-

n

, we conclude that

y

n

≦

g

≦

y

-

n

, so

g

∈

F

n

. Hence

g

∈

F

.

Note that

F

is not trivial since

y

∉

F

. The reason for this is that

x

∉

F

n

for any

n

because we assumed that

x

<

y

n

for all

n

. Hence, the order of the group

G

must be at least 2 because

F

and

{

1

}

are two examples of isolated subgroups of

F

.

Q.E.D.

Title

characterization of ordered groups of rank one

Canonical name

CharacterizationOfOrderedGroupsOfRankOne

Date of creation

2013-03-22 14:55:15

Last modified on

2013-03-22 14:55:15

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 06A05

Classification

msc 20F60

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