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Is it true that $\mathbb{Z}^{2}*\mathbb{Z}^{2}$ has a group presentation: $\langle a,b,c,d|ab=ba,cd=dc\rangle$. If not, what is the correct group presentation. Thanks!

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Yes, it is true. – Andrea May 19 '12 at 15:07
@Andrea: maybe you should post that as an answer? (perhaps with some notes why) – Ben Millwood May 19 '12 at 17:54
up vote 1 down vote accepted

Let $\,\{a,b\}\,\,,\,\,\{c,d\}\,$ be the (free abelian) generators of the first and second factors, resp., in the free product. Clearly, $\,ab=ba\,\,,\,\,cd=dc\,$ are relations in each of these factors resp. and, thus, also in the free product, and there can't be any more relations as both factors are free abelian groups in those generators.

You may want to check theorem 11.53 in Rotman's "An Introduction to the Theory of Groups", 4th Ed.

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