Quotient-transitivity and cyclic subgroup-transitivity

Brendan Goldsmith 1  and Ketao Gong 2
  • 1 School of Mathematical Sciences, TU Dublin, City Campus, Lower Grangegorman, Dublin, Ireland
  • 2 School of Mathematics and Statistics, Hubei Engineering University, No. 272, Jiaotong Road, Xiaogan City, P. R. China
Brendan Goldsmith and Ketao Gong
  • Corresponding author
  • School of Mathematics and Statistics, Hubei Engineering University, No. 272, Jiaotong Road, Xiaogan City, Hubei Province, 432000, P. R. China
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Abstract

We introduce two new notions of transitivity for Abelian 𝑝-groups based on isomorphism of quotients rather than the classical use of equality of height sequences associated with Abelian 𝑝-group theory. Unlike the classical theory where “most” groups are transitive, these new notions lead to much smaller classes, but even these classes are sufficiently large to be interesting.

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