# Variety-containing implies omega subgroup in group of prime power order

From Groupprops

Revision as of 22:33, 10 August 2009 by Vipul (talk | contribs) (Created page with '==Definition== Suppose <math>P</math> is a group of prime power order, i.e., a finite <math>p</math>-group for some prime number <math>p</math>. Suppose <math>H</math> i…')

## Definition

Suppose is a group of prime power order, i.e., a finite -group for some prime number . Suppose is a Subisomorph-containing subgroup (?) of : a subgroup of such that any subgroup of isomorphic to a subgroup of is itself contained in . Then, is one of the omega subgroups of . In other words, there is some such that:

.

## Proof

**Given**: A finite -group , a subisomorph-containing subgroup of .

**To prove**: for some natural number .

**Proof**: Let be the exponent of . Then, we clearly have:

.

Next, we show that . Suppose is such that . Then, since is the exponent of , there exists such that the order of is . Suppose the order of is , . Then, . Thus, is isomorphic to a subgroup of , so . Thus, .