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Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Cyclic groupZn Symmetric groupSn Alternating groupAn Dihedral groupDn Quaternion groupQ Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
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Ingroup theory, a branch of mathematics, given a group G under a binary operation ∗, a subset HofG is called a subgroupofGifH 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]

Aproper subgroup of a group G is a subgroup H which is a proper subsetofG (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]

IfH is a subgroup of G, then G is sometimes called an overgroupofH.

The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups.

Subgroup tests[edit]

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 binH, the product ab is in H. Closed under inverses means that for every ainH, the inverse a⁻¹ is in H. These two conditions can be combined into one, that for every a and binH, 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 Hisfinite, 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 aofH generates a finite cyclic subgroup of H, say of order n, and then the inverse of aisaⁿ⁻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 binH, the sum a + b is in H, and closed under inverses should be edited to say that for every ainH, the inverse −a is in H.

Basic properties of subgroups[edit]

• 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.
• IfH is a subgroup of G, then the inclusion map H → G sending each element aofH to itself is a homomorphism.
• The intersection of subgroups A and BofG 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 ⊆ BorB ⊆ 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 ${\displaystyle \mathbb {R} ^{2}.}$
• IfS 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⟩isisomorphicto${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ (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 orderofa. 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[edit]

Given a subgroup H and some ainG, we define the left coset aH = {ah : hinH}. 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 indexofHinG 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 ordersofG and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisorof|G|.^[7][8]

Right cosets are defined analogously: Ha = {ha : hinH}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H].

IfaH = Ha for every ainG, 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[edit]

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 tableis

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 Giscyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.^[9]

Example: Subgroups of S₄[edit]

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.

All 30 subgroups Simplified Hasse diagrams of the lattice of subgroupsofS4

12 elements[edit]

The alternating group contains only the even permutations.
It is one of the two nontrivial proper normal subgroupsofS₄. (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.

Other examples[edit]

• 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.
• Anideal in a ring R is a subgroup of the additive group of R.
• Alinear 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.

See also[edit]

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

Notes[edit]

References[edit]

• Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1.
• Hungerford, Thomas (1974), Algebra (1st ed.), Springer-Verlag, ISBN 9780387905181.
• Artin, Michael (2011), Algebra (2nd ed.), Prentice Hall, ISBN 9780132413770.
• Dummit, David S.; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 9780471452348. OCLC 248917264.
• Gallian, Joseph A. (2013). Contemporary abstract algebra (8th ed.). Boston, MA: Brooks/Cole Cengage Learning. ISBN 978-1-133-59970-8. OCLC 807255720.
• Kurzweil, Hans; Stellmacher, Bernd (1998). Theorie der endlichen Gruppen. Springer-Lehrbuch. doi:10.1007/978-3-642-58816-7.
• Ash, Robert B. (2002). Abstract Algebra: The Basic Graduate Year. Department of Mathematics University of Illinois.