Odd-order elementary abelian group is fully invariant in holomorph

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This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
View other such facts for p-groups|View other such facts for finite groups


Let p be an odd prime and consider the Elementary abelian group (?) G = C_p \times C_p \times \dots \times C_p with the product taken r times. Then, G is a Fully invariant subgroup (?) inside its holomorph.

Facts used

  1. Additive group of a field implies monolith in holomorph
  2. Monolith is fully invariant in co-Hopfian group (and in particular, in a finite group).


The proof follows from facts (1) and (2), along with the observations that: (i) any elementary abelian group is the additive group of a field, so we can use (1), and (ii) the holomorph of a finite group is again finite, so we can use (2).