Iterated agemo subgroup not implies agemo subgroup

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., iterated agemo subgroup of group of prime power order) need not satisfy the second subgroup property (i.e., agemo subgroup of group of prime power order)
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EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property iterated agemo subgroup of group of prime power order but not agemo subgroup of group of prime power order|View examples of subgroups satisfying property iterated agemo subgroup of group of prime power order and agemo subgroup of group of prime power order

Statement

It is possible to have a group of prime power order G and an iterated agemo subgroup H of G such that H is not an agemo subgroup of G.

Proof

Example for p = 2

Further information: SmallGroup(64,34), SmallGroup(64,35)

Let G be the group that we can construct as SmallGroup(64,34) using GAP. Then, we have \mho^1(G) is a group of order 16 isomorphic to the direct product of D8 and Z2, and \mho^2(G) is trivial because G has exponent 4.

The group \mho^1(\mho^1(G)) is a cyclic subgroup of order two, which is thus an iterated agemo subgroup not equal to any of the agemo subgroups.

The same observations hold for SmallGroup(64,35).

GAP implementation of proof

gap> G := SmallGroup(64,34);
<pc group of size 64 with 6 generators>
gap> IdGroup(Agemo(G,2,1));
[ 16, 11 ]
gap> G := SmallGroup(64,34);
<pc group of size 64 with 6 generators>
gap> H := Agemo(G,2,1);
Group([ f3, f4, f5, f6 ])
gap> K := Agemo(H,2,1);
Group([ f6 ])
gap> L := Agemo(G,2,2);
Group([ <identity> of ... ])
gap> IsTrivial(L);
true
gap> IdGroup(K);
[ 2, 1 ]
gap> IdGroup(H);
[ 16, 11 ]