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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).^{[1]}^{[2]}
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.
Basic properties of subgroups
 A subset H of the group G is a subgroup of 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.^{[3]} 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.
Cosets and Lagrange's theorem
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 a_{1} ~ a_{2} if and only if a_{1}^{−1}a_{2} is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H].
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.
Right cosets are defined analogously: Ha = {ha : h in H}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H].
If aH = Ha for every a in G, then H is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.
Example: Subgroups of Z_{8}
Let G be the cyclic group Z_{8} whose elements are
and whose group operation is addition modulo eight. Its Cayley table is
+  0  2  4  6  1  3  5  7 

0  0  2  4  6  1  3  5  7 
2  2  4  6  0  3  5  7  1 
4  4  6  0  2  5  7  1  3 
6  6  0  2  4  7  1  3  5 
1  1  3  5  7  2  4  6  0 
3  3  5  7  1  4  6  0  2 
5  5  7  1  3  6  0  2  4 
7  7  1  3  5  0  2  4  6 
This group has two nontrivial subgroups: J={0,4} and H={0,2,4,6}, where J is also a subgroup of H. The Cayley table for H is the topleft quadrant of the Cayley table for G. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
Example: Subgroups of S_{4 }(the symmetric group on 4 elements)
Every group has as many small subgroups as neutral elements on the main diagonal:
The trivial group and twoelement groups Z_{2}. These small subgroups are not counted in the following list.

12 elements
8 elements
  
6 elements
   
4 elements
   
  
3 elements
  

Other examples
 An ideal in a ring is a subgroup of the additive group of .
 Let be an abelian group; the elements of that have finite period form a subgroup of called the torsion subgroup of .
See also
Notes
References
 Jacobson, Nathan (2009), Basic algebra, 1 (2nd ed.), Dover, ISBN 9780486471891.
 Hungerford, Thomas (1974), Algebra (1st ed.), SpringerVerlag, ISBN 9780387905181.
 Artin, Michael (2011), Algebra (2nd ed.), Prentice Hall, ISBN 9780132413770.