2-groups with an odd-order automorphism that is the identity by Mazurov V.D.

By Mazurov V.D.

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Additional info for 2-groups with an odd-order automorphism that is the identity on involutions

Sample text

1. 31 Our general notation mainly concerns group theory — our standard reference being [28] — and homological algebra — our standard reference being [18]. In particular, if G is a ﬁnite group, recall that Op (G) , Op (G) , Op (G) and Op (G) respectively denote the minimal or the maximal normal subgroups of G with their index or their order being a power of p or prime ˆ except to p ; note that this notation still makes sense for a ﬁnite k ∗ -group G ˆ that Op (G) remains a p-group. For any pair of subgroups H and K of G , we denote by TG (K, H) the set of x ∈ G fulﬁlling xKx−1 ⊂ H .

But, as a matter of fact, when dealing with contravariant functors a from F to Ab (cf.

3), they coincide (cf. 4). 7 F(b,G) is a Frobenius P -category. 1; thus, since p does not divide |NG (P, e)/P ·CG (P )| (cf. 1. Let Q be a subgroup of P , K a subgroup of Aut(Q) containing FQ (Q) and ϕ : Q → P an F(b,G) -morphism such that ϕ(Q) is fully ϕK-normalized in F(b,G) ; in particular, denoting by f the block of CG (Q) such that (P, e) contains (Q, f ) , there is x ∈ G fulﬁlling (cf. 2. As above, denote by g the block of CG NPK (Q) such that (P, e) contains the Brauer (b, G)-pair (NPK (Q), g) .