In this paper we prove that every nonabelian finite 2-group with a cyclic commutator subgroup has a noninner automorphism of order two fixing either Φ(G) or Z(G) elementwise. This, together with a result of Peter Schmid on regular p-groups, extends our result to the class of nonabelian finite p-groups with a cyclic commutator subgroup.2010 Mathematics subject classification: primary 20D45; secondary 20D15.

In this paper, for any integers m 2, n 3, we exhibit a non-abelian n-generator group of order 2 2nmÀ2 with exponent 2 m whose automorphism group is isomorphic to Z n 2 2 Â Z 2 mÀ2 .

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SynopsisIn this paper we study a class of 2-generator 2-relator groups G(m) and show that they are all finite. Moreover, two infinite subclasses are soluble of derived length 4.

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