Let be a finite group admitting a coprime automorphism . Let denote the set of all commutators , where belongs to an ‐invariant Sylow subgroup of . We show that is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set is soluble or nilpotent, respectively.
Coprime automorphisms of finite groups
Let G G be a finite group admitting a coprime automorphism α \alpha of order e e . Denote by I G ( α ) I_G(\alpha ) the set of commutators g − 1 g α g^{-1}g^\alpha , where g ∈ G g\in G , and by [ G , α ] [G,\alpha ] the subgroup generated by I G ( α ) I_G(\alpha ) . We study the impact of I G ( α ) I_G(\alpha ) on the structure of [ G , α ] [G,\alpha ] . Suppose that each subgroup generated by a subset of I G ( α ) I_G(\alpha ) can be generated by at most r r elements. We show that the rank of [ G , α ] [G,\alpha ] is ( e , r ) (e,r) -bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of I G ( α ) I_G(\alpha ) has odd order, then [ G , α ] [G,\alpha ] has odd order too. Further, if every pair of elements from I G ( α ) I_G(\alpha ) generates a soluble, or nilpotent, subgroup, then [ G , α ] [G,\alpha ] is soluble, or respectively nilpotent.
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- Award ID(s):
- 1901595
- PAR ID:
- 10415803
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society
- Volume:
- 375
- Issue:
- 1058
- ISSN:
- 0002-9947
- Page Range / eLocation ID:
- 4549 to 4565
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/tran/8553
