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 cyclic group Z₈ whose elements are

${\displaystyle G=\left\{0,4,2,6,1,5,3,7\right\}}$ 

and whose group operation is addition modulo 8. Its Cayley table is

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]

S₄ 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.

24 elements edit

Like each group, S₄ is a subgroup of itself.

12 elements edit

The alternating group contains only the even permutations.
It is one of the two nontrivial proper normal subgroups of S₄. (The other one is its Klein subgroup.)

8 elements edit

6 elements edit

4 elements edit

3 elements edit

2 elements edit

Each permutation p of order 2 generates a subgroup {1, p}. These are the permutations that have only 2-cycles:

• There are the 6 transpositions with one 2-cycle.   (green background)
• And 3 permutations with two 2-cycles.   (white background, bold numbers)

1 element edit

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.
• An ideal in a ring R is a subgroup of the additive group of R.
• A 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.

• Cartan subgroup
• Fitting subgroup
• Fixed-point subgroup
• Fully normalized subgroup
• Stable subgroup

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• Hungerford, Thomas (1974), Algebra (1st ed.), Springer-Verlag, ISBN 9780387905181.
• Artin, Michael (2011), Algebra (2nd ed.), Prentice Hall, ISBN 9780132413770.
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• Gallian, Joseph A. (2013). Contemporary abstract algebra (8th ed.). Boston, MA: Brooks/Cole Cengage Learning. ISBN 978-1-133-59970-8. OCLC 807255720.
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