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01274nam a2200157Ia 4500 |
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10.1090-tran-8553 |
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220630s2022 CNT 000 0 und d |
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|a 00029947 (ISSN)
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| 245 |
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|a COPRIME AUTOMORPHISMS OF FINITE GROUPS
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| 260 |
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|b American Mathematical Society
|c 2022
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|a Let G be a finite group admitting a coprime automorphism α of order e. Denote by IG(α) the set of commutators g−1gα, where g ∈ G, and by [G, α] the subgroup generated by IG(α). We study the impact of IG(α) on the structure of [G, α]. Suppose that each subgroup generated by a subset of IG(α) can be generated by at most r elements. We show that the rank of [G, α] is (e, r)-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of IG(α) has odd order, then [G, α] has odd order too. Further, if every pair of elements from IG(α) generates a soluble, or nilpotent, subgroup, then [G, α] is soluble, or respectively nilpotent. © 2022 American Mathematical Society
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|a Acciarri, C.
|e author
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|a Guralnick, R.M.
|e author
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|a Shumyatsky, P.
|e author
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| 773 |
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|t Transactions of the American Mathematical Society
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| 856 |
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|z View Fulltext in Publisher
|u https://doi.org/10.1090/tran/8553
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