free product of groups

Given groups $G_i$, $i\in I$, their **free product** $G_1 \star G_2 \star \ldots = \star_i G_i$ is their coproduct in Grp.

Given presentations for the $G_i$, it is straightforward to find a presentation of $\star_i G_i$; if $G_i = \langle S_i|R_i\rangle = F_i/N_i$, where each $F_i=\langle S_i\rangle$ is the free group on the set $S_i$ and $N_i\subset F_i$ is the normal subgroup of $F_i$ generated by the subset $R_i\subset F_i$, then the free product

$\star_i G_i := \langle \coprod_i S_i | \coprod_i R_i \rangle = (\star_i F_i)/\langle\cup_i N_i\rangle$

is presented by the disjoint unions of the $S_i$ and the $R_i$. As with anything satisfying a universal property, the result (up to a unique coherent isomorphism) does not depend on the presentations.

The fact that free products always exist now follows from the fact that any group has a presentation; we can always take $S_i$ to be the underlying set of $G_i$ and take $R_i$ to be the set of all words in $F_i$ that equal the identity in $G_i$. The value of the more general construction above is that one often has much smaller $S_i$ and $R_i$ to work with. Even if $G_i$ is infinite, the $S_i$ and $R_i$ might be finite (in the strictest sense), making this part of finite mathematics and directly subject to the methods of combinatorial group theory.

Last revised on October 8, 2010 at 15:17:46. See the history of this page for a list of all contributions to it.