Variety-containing implies omega subgroup in group of prime power order
Suppose is a group of prime power order, i.e., a finite -group for some prime number . Suppose is a Variety-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 . More specifically, if the exponent of is , then:
Note that by the equivalence of definitions of variety-containing subgroup of finite group, assuming that is a variety-containing subgroup of is equivalent to assuming that it is a subhomomorph-containing subgroup or that it is a subisomorph-containing subgroup.
Given: A finite -group , a variety-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, .