Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted H ≤ G, read as 'H is a subgroup of G'. In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted H ≤ G, read as 'H is a subgroup of G'. The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (i. e. H ≠ G). This is usually represented notationally by H < G, read as 'H is a proper subgroup of G'. Some authors also exclude the trivial group from being proper (i. e. {e} ≠ H ≠ G). If H is a subgroup of G, then G is sometimes called an overgroup of H. The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures. This article will write ab for a ∗ b, as is usual. Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}. Because a is invertible, the map φ : H → aH given by φ(h) = ah is a bijection. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation a1 ~ a2 if and only if a1−1a2 is in H. The number of left cosets of H is called the index of H in G and is denoted by . Lagrange's theorem states that for a finite group G and a subgroup H, where |G| and |H| denote the orders of G and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisor of |G|.