In group theory, a branch of mathematics, given a groupG under a binary operation ∗, a subsetH 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 often 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.^{[1]}
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).^{[2]}^{[3]}
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.
Subgroup tests
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Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition.
Then H is a subgroup of Gif and only ifH is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H. Closed under inverses means that for every a in H, the inverse a^{−1} is in H. These two conditions can be combined into one, that for every a and b in H, the element ab^{−1} is in H, but it is more natural and usually just as easy to test the two closure conditions separately.^{[4]}
When H is finite, the test can be simplified: H is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element a of H generates a finite cyclic subgroup of H, say of order n, and then the inverse of a is a^{n−1}.^{[4]}
If the group operation is instead denoted by addition, then closed under products should be replaced by closed under addition, which is the condition that for every a and b in H, the sum a + b is in H, and closed under inverses should be edited to say that for every a in H, the inverse −a is in H.
Basic properties of subgroups
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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}.
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.^{[5]} For example, the intersection of the x-axis and y-axis in $\mathbb {R} ^{2}$ 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: $2\mathbb {Z} \cup 3\mathbb {Z}$ is not a subgroup of $\mathbb {Z} ,$ 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 $\mathbb {R} ^{2}$ is not a subgroup of $\mathbb {R} ^{2}.$
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 ⟨S⟩ and is called 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, possibly repeated.^{[6]}
Every element a of a group G generates a cyclic subgroup ⟨a⟩. If ⟨a⟩ is isomorphic to $\mathbb {Z} /n\mathbb {Z}$ (the integers mod n) 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 $\mathbb {Z} ,$ 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 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
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Given a subgroup H and some a in G, we define the left cosetaH = {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 relationa_{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].
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|.^{[7]}^{[8]}
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.
This group has two nontrivial subgroups: ■J = {0, 4} and ■H = {0, 4, 2, 6} , where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G; The Cayley table for J is the top-left quadrant of the Cayley table for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.^{[9]}
Example: Subgroups of S_{4}
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S_{4} is the symmetric group whose elements correspond to the permutations of 4 elements.
Below are all its subgroups, ordered by cardinality.
Each group (except those of cardinality 1 and 2) is represented by its Cayley table.
The even integers form a subgroup $2\mathbb {Z}$ of the integer ring$\mathbb {Z} :$ the sum of two even integers is even, and the negative of an even integer is even.
An ideal in a ring R is a subgroup of the additive group of R.