Exercise 35, § 6

Algebra, I, 156

Let G be a group such that the derived group D = D(G) is contained in the centre C of G.

(a) Show that the mapping $(x, y) \mapsto (x, y)$ induces a mapping

\phi : (G/C) \times (G/C) \to D

such that, writing these groups additively:

\begin{align*} \phi(\alpha + \beta, \gamma) &= \phi(\alpha, \gamma) + \phi(\beta, \gamma) & \phi(\alpha, \beta) &= -\phi(\beta, \alpha) \\ \phi(\alpha, \beta + \gamma) &= \phi(\alpha, \beta) + \phi(\alpha, \gamma) & \phi(\alpha, \alpha) &= 0 \end{align*}

for all

\alpha, \beta, \gamma \in G/C

(b) Show that, for every integer n,

(yx)^n = y^n x^n (x, y)^{n(n-1)/2}

for

x, y \in G

. Deduce that if n is odd and

d^n = 1

for all

d \in D

, the mapping

x \mapsto x^n

induces a homomorphism

\theta : G/D \to C

Solution

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