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A long-standing conjecture asserts that every finite nonabelian $p$-group has a noninner automorphism of order $p$. In this paper the verification of the conjecture is reduced to the case of $p$-groups $G$ satisfying ${ Z}_{2}^{\star } (G)\leq {C}_{G} ({ Z}_{2}^{\star } (G))= \Phi (G)$, where ${ Z}_{2}^{\star } (G)$ is the preimage of ${\Omega }_{1} ({Z}_{2} (G)/ Z(G))$ in $G$. This improves Deaconescu and Silberberg’s reduction of the conjecture: if ${C}_{G} (Z(\Phi (G)))\not = \Phi (G)$, then $G$ has a noninner automorphism of order $p$ leaving the Frattini subgroup of $G$ elementwise fixed [‘Noninner automorphisms of order $p$ of finite $p$-groups’, J. Algebra 250 (2002), 283–287].

Let p be an odd prime and let G be a finite p-group such that xZ(G)⊆xG, for all x∈G∖Z(G), where xG denotes the conjugacy class of x in G. Then G has a noninner automorphism of order p leaving the Frattini subgroup Φ(G) elementwise fixed.

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NONINNER AUTOMORPHISMS OF ORDER $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ IN FINITE $p$-GROUPS OF COCLASS 2, WHEN $p>2$

ON NONINNER AUTOMORPHISMS OF FINITE NONABELIAN $p$-GROUPS

A NOTE ON AUTOMORPHISMS OF FINITE p-GROUPS

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3 results

NONINNER AUTOMORPHISMS OF ORDER $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ IN FINITE $p$-GROUPS OF COCLASS 2, WHEN $p>2$

ON NONINNER AUTOMORPHISMS OF FINITE NONABELIAN $p$-GROUPS

A NOTE ON AUTOMORPHISMS OF FINITE p-GROUPS