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In the mathematical subject known as group theory, given a group G under a binary operation *, we say that some subset H of G is 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 x H is a group operation on H. This is usually represented notationally by H ≤ G, read as "H is a subgroup of G".
A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. H ≠ G). The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. 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.
In the following, we follow the usual convention of dropping * and writing the product a*b as simply ab.
Contents
Basic properties of subgroups
 H is a subgroup of the group G if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever a and b are in H, then ab and a^{−1} are also in H. These two conditions can be combined into one equivalent condition: whenever a and b are in H, then ab^{−1} is also in H.) In the case that H is finite, then H is a subgroup if and only if H is closed under products. (In this case, every element a of H generates a finite cyclic subgroup of H, and the inverse of a is then a^{−1} = a^{n − 1}, where n is the order of a.)
 The above condition can be stated in terms of a homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism (i.e., i(a) = a for every a) from H to G.
 The identity of a subgroup is the identity of the group: if G is a group with identity e_{G}, and H is a subgroup of G with identity e_{H}, then e_{H} = e_{G}.
 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 = e_{H}, then ab = ba = e_{G}.
 The intersection of subgroups A and B is again a subgroup.^{[1]} The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the xaxis and the yaxis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
 If S is a subset of G, then there exists a minimum subgroup containing S, which can be found by taking the intersection of all of subgroups containing S; it is denoted by <S> and is said to be the subgroup generated by S. An element of G is in <S> if and only if it is a finite product of elements of S and their inverses.
 Every element a of a group G generates the cyclic subgroup <a>. If <a> is isomorphic to Z/nZ for some positive integer n, then n is the smallest positive integer for which a^{n} = e, and n is called the order of a. If <a> 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 settheoretic intersection, the supremum of a set of subgroups is the subgroup generated by the settheoretic union of the subgroups, not the settheoretic union itself.) If e is the identity of G, then the trivial group {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself.
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