I am encountering free products for the first time in Algebraic Topology during the discussion of van Kampen's theorem, and I can't seem to tell the difference between a free product of groups, and a free group. The definition I know (of a free product) is:
Let $G,H$ be groups. Define $G *H$ to be the set of all formal words $g_1h_1 \cdots g_nh_k$ where $g_i \in G$ and $h_j \in H$. Then $G*H$ is a group under the operation of juxtaposition, and the identity is the empty word.
This definition can be extended to the free product of an arbitrary collection $G_\alpha$ of groups, but I don't see how this definition is different from the free group on a set $S$. My guesses at the differences:
- Free groups can be formed with any set $S$, whereas free products are defined for collections of groups.
- Free products respects relations between the groups. For example, in $G*H$ the word $g_1g_2h_1$ is just $g_3h_1$ for some $g_3=g_1g_2 \in G$. Whereas there is no underlying relation for words in a free group (except for the formal cancellation of words).
And one last questions: are all free groups also free products? Or is the inclusion in the other direction?