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In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.

Formally, 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 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.

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 G if and only if H 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 aⁿ⁻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.

• 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 ⁠${\displaystyle \mathbb {R} ^{2}}$⁠ 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: ⁠${\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} }$⁠ is not a subgroup of ⁠${\displaystyle \mathbb {Z} ,}$⁠ 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 ⁠${\displaystyle \mathbb {R} ^{2}}$⁠ is not a subgroup of ⁠${\displaystyle \mathbb {R} ^{2}.}$⁠
• 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 ⁠${\displaystyle \mathbb {Z} /n\mathbb {Z} }$⁠ (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 ⟨a⟩ is isomorphic to ⁠${\displaystyle \mathbb {Z} ,}$⁠ 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.

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 ⁠${\displaystyle a_{1}^{-1}a_{2}}$⁠ 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,

${\displaystyle [G:H]={|G| \over |H|}}$

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.

Let G be the finite cyclic group

${\displaystyle \mathrm {Z} _{8}=\{0,1,2,3,4,5,6,7\}}$

under addition modulo 8. The subset ${\displaystyle \{0,2,4,6\}}$ consisting of multiples of 2 is a subgroup of ${\displaystyle \mathrm {Z} _{8}}$. More generally, for each divisor d of 8, the multiples of d form a subgroup. Explicitly, for ${\displaystyle d=1,2,4,8}$, these subgroups are ${\displaystyle \{0,1,2,3,4,5,6,7\},\{0,2,4,6\},\{0,4\},\{0\}}$.

In general, for any positive integer n, one can describe all subgroups of the finite cyclic group ${\displaystyle \mathrm {Z} _{n}}$ similarly: for each divisor d of n, the multiples of d in ${\displaystyle \mathrm {Z} _{n}}$ form a subgroup of order ${\displaystyle n/d}$, and every subgroup arises in this way.

Subgroups of cyclic groups are cyclic.^[9]

The symmetric group S₄ is the group whose elements are the permutations of ${\displaystyle \{1,2,3,4\}}$.
Below are all its subgroups, ordered by cardinality.

All 30 subgroups Simplified Hasse diagrams of the lattice of subgroups of S4

Like each group, S₄ is a subgroup of itself.

The alternating group A₄ consists of all the even permutations in S₄. Since it is of index 2, it is a normal subgroup.

There are three subgroups of order 8, each isomorphic to the dihedral group D₄, the group of symmetries of a square.

Labeling the vertices of a square ${\displaystyle 1,2,3,4}$ clockwise lets one view D₄ as a subgroup of S₄. This subgroup is generated by the 90-degree clockwise rotation and by the reflection in the diagonal axis joining vertices 1 and 3; these are the permutations ${\displaystyle (1234)}$ and ${\displaystyle (24)}$.

Up to symmetries of the square, there are three different ways to label the vertices of a square, distinguished by which pairs of numbers appear on opposite corners. In the labeling above, 1 and 3 were opposite, and 2 and 4 were opposite; another choice has 1 and 4 opposite, and 2 and 3 opposite; the third choice has 1 and 2 opposite, and 3 and 4 opposite. The three labelings give rise to three different subgroups of order 8 in S₄, conjugate to each other, each isomorphic to D₄.

There are four subgroups of order 6, each isomorphic to S₃. Each is the stabilizer of one of the elements of ${\displaystyle \{1,2,3,4\}}$. For example, the stabilizer of 4 is the group of permutations in S₄ that map 4 to 4, while permuting ${\displaystyle \{1,2,3\}}$ in an arbitrary way; it is generated by the permutations ${\displaystyle (12)}$ and ${\displaystyle (123)}$, for instance. The four subgroups of order 6 are conjugate to each other.

There are seven subgroups of order 4, falling into three conjugacy classes of subgroups:

• The subset ${\displaystyle \{1,(12)(34),(13)(24),(14)(23)\}}$ is a normal subgroup isomorphic to the Klein four-group V₄.

• The group generated by ${\displaystyle (12)}$ and ${\displaystyle (34)}$ is another subgroup isomorphic to V₄, but it is not normal. Instead it has conjugates, namely the group generated by ${\displaystyle (13)}$ and ${\displaystyle (24)}$ and the group generated by ${\displaystyle (14)}$ and ${\displaystyle (23)}$.

• Each of the six 4-cycles in S₄ generates a cyclic subgroup of order 4, but each 4-cycle generates the same subgroup as its inverse, so there are only three distinct subgroups of this type. These three subgroups are conjugate to each other because all 4-cycles in S₄ are conjugate to each other.

There are four subgroups of order 3, each generated by a 3-cycle. There are eight 3-cycles in S₄, but each generates the same subgroup as its inverse. The resulting four subgroups are conjugate to each other.

There are nine subgroups of order 2, falling into two conjugacy classes of subgroups:

• Each of the ${\displaystyle {\binom {4}{2}}=6}$ transpositions (2-cycles) generates a subgroup of order 2. These six subgroups are conjugate.

• Each of the double-transpositions ${\displaystyle (12)(34)}$, ${\displaystyle (13)(24)}$, ${\displaystyle (14)(23)}$ generates a subgroup of order 2. These three subgroups are conjugate.

The trivial subgroup is the unique subgroup of order 1.

• The even integers form a subgroup ⁠${\displaystyle 2\mathbb {Z} }$⁠ of the integer ring ⁠${\displaystyle \mathbb {Z} :}$⁠ the sum of two even integers is even, and the negative of an even integer is even.
• Every ideal in a ring R is a subgroup of the additive group of R.
• Every linear subspace of a vector space is a subgroup of the additive group of vectors.
• In an abelian group, the elements of finite order form a subgroup called the torsion subgroup.

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