For other uses, see Subgroup (disambiguation).

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 eG, and H is a subgroup of G with identity eH, then eH = eG.
  • 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 = eH, then ab = ba = eG.
  • 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 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 \cup 3\Z is not a subgroup of \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 is not a subgroup of \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 and is called the subgroup generated by S. An element of G is in 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 . If is isomorphic to \Z/n\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 is isomorphic to \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

Main articles: Coset and Lagrange's theorem (group theory)

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 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| \over |H| }

where and 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 .[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

\mathrm{Z}_8 = \{0,1,2,3,4,5,6,7\}

under addition modulo 8. The subset \{0,2,4,6\} consisting of multiples of 2 is a subgroup of \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, these subgroups are \{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 \mathrm{Z}_n similarly: for each divisor d of n, the multiples of d in \mathrm{Z}_n form a subgroup of order 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\}.
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 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 (1234) and (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 \{1,2,3,4\}. For example, the stabilizer of 4 is the group of permutations in S4 that map 4 to 4, while permuting \{1,2,3\} in an arbitrary way; it is generated by the permutations (12) and (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 \{1,(12)(34),(13)(24),(14)(23)\} is a normal subgroup isomorphic to the Klein four-group V4.
  • The group generated by (12) and (34) is another subgroup isomorphic to V4, but it is not normal. Instead it has conjugates, namely the group generated by (13) and (24) and the group generated by (14) and (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 \binom{4}{2} = 6 transpositions (2-cycles) generates a subgroup of order 2. These six subgroups are conjugate.
  • Each of the double-transpositions (12)(34), (13)(24), (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 2\Z of the integer ring \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

  1. ^ Gallian 2013, p. 61.
  2. ^ Hungerford 1974, p. 32.
  3. ^ Artin 2011, p. 43.
  4. ^ Kurzweil & Stellmacher 1998, p. 4.
  5. ^ Jacobson 2009, p. 41.
  6. ^ Ash 2002.
  7. ^ See a didactic proof in this video.
  8. ^ Dummit & Foote 2004, p. 90.
  9. ^ Gallian 2013, p. 81.

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.