Exercise 35, § 6

Algebra, I, p. 140

Let $G$ be a group such that the derived group $D = D(G)$ is contained in the center C of G. a) Prove that the mapping $(x, y) \mapsto (x, y)$ induces a mapping $\varphi : (G/C) \times (G/C) \to D$ such that, on writing these groups additively, one has:

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

for all

\alpha, \beta, \gamma

G/C

. b) Prove that, for every integer

n

, one has

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

for $x, y$ in $G$ . Deduce that, if $n$ is odd and if $d^n = 1$ for every $d \in D$ , the mapping $x \mapsto x^n$ defines, by passing to the quotient, a homomorphism $\theta : G/D \to C$ .

Solution

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