nLab free product of groups

Given groups G iG_i, iIi\in I, their free product G 1G 2= iG iG_1 \star G_2 \star \ldots = \star_i G_i is their coproduct in Grp.

Given presentations for the G iG_i, it is straightforward to find a presentation of iG i\star_i G_i; if G i=S i|R i=F i/N iG_i = \langle S_i|R_i\rangle = F_i/N_i, where each F i=S iF_i=\langle S_i\rangle is the free group on the set S iS_i and N iF iN_i\subset F_i is the normal subgroup of F iF_i generated by the subset R iF iR_i\subset F_i, then the free product

iG i:= iS i| iR i=( iF i)/ iN i\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 iS_i and the R iR_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 iS_i to be the underlying set of G iG_i and take R iR_i to be the set of all words in F iF_i that equal the identity in G iG_i. The value of the more general construction above is that one often has much smaller S iS_i and R iR_i to work with. Even if G iG_i is infinite, the S iS_i and R iR_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.