Subgroups

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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.

Subgroup tests

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.

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.^[5] For example, the intersection of the x-axis and y-axis in ⁠ R 2 {\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: ⁠ 2 Z ∪ 3 Z {\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} } ⁠ is not a subgroup of ⁠ Z , {\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 ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠ is not a subgroup of ⁠ R 2 . {\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 ⁠ Z / n Z {\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 ⁠ Z , {\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 ⁠ a 1 − 1 a 2 {\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,

[ G : H ] = | G | | 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.

Example: Subgroups of Z₈

Let G be the finite cyclic group

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

under addition modulo 8. The subset { 0 , 2 , 4 , 6 } {\displaystyle \{0,2,4,6\}} consisting of multiples of 2 is a subgroup of Z 8 {\displaystyle \mathrm {Z} _{8}} . More generally, for each divisor d of 8, the multiples of d form a subgroup. Explicitly, for d = 1 , 2 , 4 , 8 {\displaystyle d=1,2,4,8} , these subgroups are { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } , { 0 , 2 , 4 , 6 } , { 0 , 4 } , { 0 } {\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 Z n {\displaystyle \mathrm {Z} _{n}} similarly: for each divisor d of n, the multiples of d in Z n {\displaystyle \mathrm {Z} _{n}} form a subgroup of order n / d {\displaystyle n/d} , and every subgroup arises in this way.

Subgroups of cyclic groups are cyclic.^[9]

Example: Subgroups of S₄

The symmetric group S₄ is the group whose elements are the permutations of { 1 , 2 , 3 , 4 } {\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

Other examples

• The even integers form a subgroup ⁠ 2 Z {\displaystyle 2\mathbb {Z} } ⁠ of the integer ring ⁠ Z : {\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.

Notes

References

• 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. ISBN 978-3-540-60331-3.
• Ash, Robert B. (2002). Abstract Algebra: The Basic Graduate Year. Department of Mathematics University of Illinois.