Exercise 16, § 6

Let V be a vector space over a commutative field k and

(cf. III, § 7, no. 1). On $E = V \times W$ a law of composition is defined by the formula

$$(v_1, w_1) \cdot (v_2, w_2) = (v_1 + v_2, w_1 + w_2 + v_1 \wedge v_2).$$

(a) Show that this law gives E a group structure, that of a central extension of V by W. (b) Suppose the characteristic of k is different from 2. Show that the derived group D(E) is the set of $(0, w), w \in W$; it is a group isomorphic to W. Show that, if $\dim(V) \geq 4$, there exist elements of D(E) which are not commutators.