# Iterated agemo subgroup not implies agemo subgroup

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., agemo subgroup of group of prime power order)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

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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 and an iterated agemo subgroup of such that is not an agemo subgroup of .

## Proof

### Example for

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

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

The group 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 ]