Subgroup

Basic properties of subgroups

• 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. 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.
• 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.

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 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,

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

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|.

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 Z8

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.

Example: Subgroups of S4

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.

24 elements

Like each group, S₄ is a subgroup of itself.

12 elements

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

8 elements

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.

6 elements

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.

4 elements

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.

3 elements

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.

2 elements

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

• Each of the ${\displaystyle (\genfrac{}{}{0ex}{}{4}{2})=6}$ 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.

1 element

The trivial subgroup is the unique subgroup of order 1.

Other examples

• 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.