Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 1 / 6 / iv / Solution

True. Let $G=⟨X\mid R⟩$ and $H=⟨Y\mid S⟩$, using disjoint finite generating alphabets. The direct product of groups has finite presentationThe commuting relations let every word be reordered as a word in $X$ followed by a word in $Y$; the two projections then show that no additional relations within either factor are imposed. For an input word $w$, form its projections $w_X$ and $w_Y$ by deleting all letters from the other factor. Evaluate each projection with the given word problem for a group algorithm for that factor. The product coordinates giveBoth component algorithms halt, so this is a terminating algorithm. No subgroup-membership procedure or comparison between unknown presentations is required.