Finite self-similar p-groups with abelian first level stabilizers

Details Finite self-similar p-groups with abelian first level stabilizers Finite self-similar p-groups with abelian first level stabilizers

2010-01-11

arxiv.org/abs/1001.1600v4

International Journal of Algebra and Computation 21 (2011) 1-2, 355--364

Mathematics

Group Theory

one direction of theorem 2 extended to regular p-groups

Scientific paper

10.1142/S0218196711006200

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian p-group by a cyclic group of order p. The proof is based on use of virtual endomorphisms. In this context the result says that if G is a finite p-group with abelian subgroup H of index p, then there exists a virtual endomorphism of G with trivial core and domain H if and only if G is a split extension of H and H is an elementary abelian p-group.

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