• Title/Summary/Keyword: cyclic maximal subgroup

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COMMUTING AUTOMORPHISM OF p-GROUPS WITH CYCLIC MAXIMAL SUBGROUPS

  • Vosooghpour, Fatemeh;Kargarian, Zeinab;Akhavan-Malayeri, Mehri
    • Communications of the Korean Mathematical Society
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    • v.28 no.4
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    • pp.643-647
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    • 2013
  • Let G be a group and let $p$ be a prime number. If the set $\mathcal{A}(G)$ of all commuting automorphisms of G forms a subgroup of Aut(G), then G is called $\mathcal{A}(G)$-group. In this paper we show that any $p$-group with cyclic maximal subgroup is an $\mathcal{A}(G)$-group. We also find the structure of the group $\mathcal{A}(G)$ and we show that $\mathcal{A}(G)=Aut_c(G)$. Moreover, we prove that for any prime $p$ and all integers $n{\geq}3$, there exists a non-abelian $\mathcal{A}(G)$-group of order $p^n$ in which $\mathcal{A}(G)=Aut_c(G)$. If $p$ > 2, then $\mathcal{A}(G)={\cong}\mathbb{Z}_p{\times}\mathbb{Z}_{p^{n-2}}$ and if $p=2$, then $\mathcal{A}(G)={\cong}\mathbb{Z}_2{\times}\mathbb{Z}_2{\times}\mathbb{Z}_{2^{n-3}}$ or $\mathbb{Z}_2{\times}\mathbb{Z}_2$.

OUTER AUTOMORPHISM GROUPS OF POLYGONAL PRODUCTS OF CERTAIN CONJUGACY SEPARABLE GROUPS

  • Kim, Goan-Su;Tang, Chi Yu
    • Journal of the Korean Mathematical Society
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    • v.45 no.6
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    • pp.1741-1752
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    • 2008
  • Grossman [7] showed that certain cyclically pinched 1-relator groups have residually finite outer automorphism groups. In this paper we prove that tree products of finitely generated free groups amalgamating maximal cyclic subgroups have residually finite outer automorphism groups. We also prove that polygonal products of finitely generated central subgroup separable groups amalgamating trivial intersecting central subgroups have residually finite outer automorphism groups.