a in H, the inverse −a is in H. Basic properties of subgroups The identity of a subgroup is the identity of the group: if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG. The inverse of an element in a subgroup is the inverse of the element in the group: if H is a subgroup of a group G, and a and b are elements of H such that ab = ba = eH, then ab = ba = eG. If H is a subgroup of G, then the inclusion map H → G sending each element a of H to itself is a homomorphism. The intersection of subgroups A and B of G is again a subgroup of G. For example, the intersection of the x-axis and y-axis in R under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G. The union of subgroups A and B is a subgroup if and only if A ⊆ B or B ⊆ A. A non-example: 2Z ∪ 3Z is not a subgroup of Z, because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the x-axis and the y-axis in R is not a subgroup of R. If S is a subset of G, then there exists a smallest subgroup containing S, namely the intersection of all of subgroups containing S; it is denoted by and is called the subgroup generated by S. An element of G is in if and only if it is a finite product of elements of S and their inverses, possibly repeated. Every element a of a group G generates a cyclic subgroup . If is isomorphic to Z/nZ (the integers mod n) for some positive integer n, then n is the smallest positive integer for which an = e, and n is called the order of a. If is isomorphic to Z, then a is said to have infinite order. The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union