Group automorphisms inducing the identity map on cohomology

Class-Preserving Automorphisms of Generalized Free Products Amalgamating a Cyclic Normal Subgroup

“…As before, we have B = ⟨b, c⟩, where It is interesting to see that the result is not true even when the amalgamating normal subgroup is of order 8 by the following example [13]. The map ϕ : x → x, y → y, z → x 4 z defines a class-preserving automorphism of A which is not inner [7]. Similarly, the map ϕ 1 : x → x, y 1 → y 1 , z 1 → x 4 z 1 defines a class-preserving automorphism of B.…”

The normalizer property for integral group rings of wreath products of finite nilpotent groups by some 2-groups

“…If this equality holds, then we say that G has the normalizer property. This equality was proved for finite nilpotent groups by Coleman [1] and later by Jackowscki and Marciniak [5] for finite groups having a normal Sylow 2-subgroup. In particular, the normalizer property holds for finite groups of odd order.…”

“…Lemma 2.5 (Jackowscki and Marciniak [5]). Let S be a fixed Sylow 2-subgroup of a finite group G and I S ¼ fj u j u A N UðZGÞ ðGÞ; j u j S ¼ id; j 2 u ¼ idg.…”

Units of p-power order in principal p-blocks of p-constrained groups

“…Coleman's contribution [1] is well known, but the first version of the lemma appears in an article of Ward [19] as a contribution to a seminar run by Richard Brauer at Harvard. See also [15, Proposition 1.14], [7,2.6 Theorem]. Actually, for its proof it is only needed that p is not invertible in the commutative ring R.…”