Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 5 / a

Solution

Let

B = ⨁_{g \in G} H_{g}

be the group of finitely supported functions

f : G \to H

(g \cdot f) (x) = f (g^{- 1} x)

H ≀ G = B ⋊ G .

Suppose

X

and

Y

are finite generating sets for

G

and

H

. Embed each

y \in Y

as a lamp supported at the identity of

G

. Conjugating these lamps by words in

X

produces copies of

Y

at every coordinate, and these copies generate

B

. Thus

X

together with the identity-coordinate copy of

Y

is a finite generating set for

H ≀ G

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