Free product

If and are groups, a word on and is a sequence of the form s_1 s_2 \cdots s_n, where each is either an element of or an element of . Such a word may be reduced using the following operations: • remove an instance of the identity element (of either or ), or • replace a pair by its product in , or a pair by its product in . Every reduced word is either the empty sequence, contains exactly one element of or , or is an alternating sequence of elements of and elements of , e.g. : g_1 h_1 g_2 h_2 \cdots g_k h_k. The free product is the group whose elements are the reduced words in and , under the operation of concatenation followed by reduction. For example, if is the infinite cyclic group \langle x\rangle, and is the infinite cyclic group \langle y\rangle, then every element of is an alternating product of powers of with powers of . In this case, is isomorphic to the free group generated by and . == Presentation ==

Suppose that : G = \langle S_G \mid R_G \rangle is a presentation for G (where SG is a set of generators and RG is a set of relations), and suppose that : H = \langle S_H \mid R_H \rangle is a presentation for H. Then : G * H = \langle S_G \cup S_H \mid R_G \cup R_H \rangle. That is, G ∗ H is generated by the generators for G together with the generators for H, with relations consisting of the relations from G together with the relations from H (assume here no notational clashes so that these are in fact disjoint unions). Examples For example, suppose that G is a cyclic group of order 4, : G = \langle x \mid x^4 = 1 \rangle, and H is a cyclic group of order 5 : H = \langle y \mid y^5 = 1 \rangle. Then G ∗ H is the infinite group : G * H = \langle x, y \mid x^4 = y^5 = 1 \rangle. Because there are no relations in a free group, the free product of free groups is always a free group. In particular, : F_m * F_n \cong F_{m+n}, where Fn denotes the free group on n generators. Another example is the modular group PSL_2(\mathbf Z). It is isomorphic to the free product of two cyclic groups: : PSL_2(\mathbf Z) \cong (\mathbf Z / 2 \mathbf Z) \ast (\mathbf Z / 3 \mathbf Z). == Generalization: Free product with amalgamation ==

The more general construction of free product with amalgamation is correspondingly a special kind of pushout in the same category. Suppose G and H are given as before, along with monomorphisms (i.e. injective group homomorphisms): : \varphi : F \rightarrow G \ \, and \ \, \psi : F \rightarrow H, where F is some arbitrary group. Start with the free product G * H and adjoin as relations : \varphi(f)\psi(f)^{-1}=1 for every f in F. In other words, take the smallest normal subgroup N of G * H containing all elements on the left-hand side of the above equation, which are tacitly being considered in G * H by means of the inclusions of G and H in their free product. The free product with amalgamation of G and H, with respect to \varphi and \psi, is the quotient group : (G * H)/N.\, The amalgamation has forced an identification between \varphi(F) in G with \psi(F) in H, element by element. This is the construction needed to compute the fundamental group of two connected spaces joined along a path-connected subspace, with F taking the role of the fundamental group of the subspace. See: Seifert–van Kampen theorem. Karrass and Solitar have given a description of the subgroups of a free product with amalgamation. For example, the homomorphisms from G and H to the quotient group (G * H)/N that are induced by \varphi and \psi are both injective, as is the induced homomorphism from F. Free products with amalgamation and a closely related notion of HNN extension are basic building blocks in Bass–Serre theory of groups acting on trees. == In other branches ==

One may similarly define free products of other algebraic structures than groups, including algebras over a field. Free products of algebras of random variables play the same role in defining "freeness" in the theory of free probability that Cartesian products play in defining statistical independence in classical probability theory. == See also ==