Number of conjugacy classes

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This article defines an arithmetic function on groups
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Definition

The number of conjugacy classes in a group is defined as the number of conjugacy classes, viz the number of equivalence classes under the equivalence relation of being conjugate.

This number is also sometimes termed the class number of the group.

Related group properties

  • A group in which every conjugacy class is finite is termed an FC-group. In particular, a FC-group is finite if and only if it has finitely many conjugacy classes.
  • A group with two conjugacy classes is a nontrivial group with exactly one conjugacy class of non-identity elements. Note that the only such finite group is cyclic group:Z2. However, there are many infinite groups with this property.

Facts

Ways of measuring this number for a finite group

Lower bounds on the number of conjugacy classes

  • The number of conjugacy classes in a group is at least as much as the number of elements in the center.
  • The number of conjugacy classes in a group is at least as much as the number of elements in the abelianization.
  • There are finitely many finite groups with bounded number of conjugacy classes. Thus, the number of conjugacy classes puts an upper bound on the order of a finite group. Conversely, the order of a group puts a lower bound on the number of conjugacy classes. On the other hand, there do exist infinite groups with only finitely many conjugacy classes.

Upper bounds on the number of conjugacy classes

  • The number of conjugacy classes is at most as much as the number of elements in the group. Equality (for finite groups) holds if and only if the group is abelian.
  • Commuting fraction more than five-eighths implies abelian: In particular, this means that for a finite non-abelian group, the number of conjugacy classes is bounded by 5/8 times the order of the group.

Relating the number of conjugacy classes for a group with subgroups, quotients, and direct products

Relationship type Bound in one direction relating number of conjugacy classes Bound in other direction relating number of conjugacy classes Non-bounds
subgroup Number of conjugacy classes in a subgroup of finite index is bounded by index times number of conjugacy classes in the whole group: This in particular puts an upper bound on the number of conjugacy classes in a subgroup in terms of the number of conjugacy classes in the whole group or equivalently a lower bound on the number of conjugacy classes in the whole group based on the number of conjugacy classes in the subgroup, using the index of the subgroup as the factor controlling the bounding. Commuting fraction in subgroup is at least as much as in whole group: This in particular implies that the quotient of the number of conjugacy classes to the order is at least as much in a subgroup as in the whole group. Thus, the number of conjugacy classes in a subgroup puts an upper bound (via multiplication by the index of the subgroup) on the number of conjugacy classes in the whole group, and the number of conjugacy classes in the whole group puts a lower bound (via division by the index of the subgroup) on the number of conjugacy classes in the subgroup. number of conjugacy classes in a subgroup may be more than in the whole group: Note that the upper and lower bounds bound each number in terms of the index of the subgroup times the other. This leaves a wide range of indeterminacy. It is possible for either of the two numbers to be bigger than the other.
quotient group number of conjugacy classes in a quotient is less than or equal to number of conjugacy classes of group: This gives an upper bound on the number of conjugacy classes in a quotient group in terms of the number of conjugacy classes in the whole group, or equivalently a lower bound on the number of conjugacy classes in the whole group based on the number of conjugacy classes in the quotient group. commuting fraction in quotient group is at least as much as in whole group: This in particular implies that the number of conjugacy classes in the whole group is at most the order of the kernel times the number of conjugacy classes in the quotient group. Thus, it gives an upper bound on the number of conjugacy classes in the whole group in terms of the quotient, and a lower bound on the number of conjugacy classes in the quotient in terms of the whole group.
external direct product number of conjugacy classes in a direct product is the product of the number of conjugacy classes in each factor, equivalently formulated as commuting fraction of direct product is product of commuting fractions same as previous column, since it's an exact formula --
group extension number of conjugacy classes in extension group is bounded by product of number of conjugacy classes in normal subgroup and quotient group

Particular cases

By number of conjugacy classes

Further information: There are finitely many finite groups with bounded number of conjugacy classes

Number of conjugacy classes List of all finite groups with that number List of orders of these groups
1 trivial group 1
2 cyclic group:Z2 2
3 cyclic group:Z3, symmetric group:S3 3, 6
4 cyclic group:Z4, Klein four-group, dihedral group:D10, alternating group:A4, more? 4, 4, 10, 12
5 cyclic group:Z5, dihedral group:D8, quaternion group, dihedral group:D14, general affine group:GA(1,5), SmallGroup(21,1), symmetric group:S4, alternating group:A5 5, 8, 8, 14, 20, 21, 24, 60

By groups

Group Order Second part of GAP ID Number of conjugacy classes Comment
trivial group 1 1 1
cyclic group:Z2 2 1 2 for an abelian group, number of conjugacy classes equals number of elements
cyclic group:Z3 3 1 3 for an abelian group, number of conjugacy classes equals number of elements
cyclic group:Z4 4 1 4 for an abelian group, number of conjugacy classes equals number of elements
cyclic group:Z5 5 1 5 for an abelian group, number of conjugacy classes equals number of elements
symmetric group:S3 6 1 3 cycle type determines conjugacy class, so number of conjugacy classes equals number of unordered integer partitions for a symmetric group. See also element structure of symmetric group:S3 and element structure of symmetric groups
cyclic group:Z6 6 2 6 for an abelian group, number of conjugacy classes equals number of elements
cyclic group:Z7 7 1 7 for an abelian group, number of conjugacy classes equals number of elements
cyclic group:Z8 8 1 8 for an abelian group, number of conjugacy classes equals number of elements
direct product of Z4 and Z2 8 2 8 for an abelian group, number of conjugacy classes equals number of elements
dihedral group:D8 8 3 5 See element structure of dihedral groups (for general formula: (n + 6)/2 for dihedral group of order 2n for even n) and element structure of dihedral group:D8 for specific information. See also nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order
quaternion group 8 4 5 See element structure of dicyclic groups (for general formula) and element structure of quaternion group for specific information. See also nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order
elementary abelian group:E8 8 5 8 for an abelian group, number of conjugacy classes equals number of elements
cyclic group:Z9 9 1 9 for an abelian group, number of conjugacy classes equals number of elements
elementary abelian group:E9 9 2 9 for an abelian group, number of conjugacy classes equals number of elements
dihedral group:D10 10 1 4 See element structure of dihedral groups (for general formula: (n + 3)/2 for dihedral group of order 2n for odd n)
cyclic group:Z10 10 2 10 for an abelian group, number of conjugacy classes equals number of elements
cyclic group:Z11 11 1 11 for an abelian group, number of conjugacy classes equals number of elements

By family

Family name Parameter value Number of conjugacy classes More information
symmetric group S_n n number of unordered integer partitions p(n) element structure of symmetric groups, cycle type determines conjugacy class
dihedral group D_{2n} n (degree, half the order) (n + 3)/2 for n odd, (n + 6)/2 for n even element structure of dihedral groups
alternating group A_n n 2A + B where A is the number of self-conjugate integer partitions of n and B = (p(n) - A)/2 is the number of conjugate pairs of non-self-conjugate unordered integer partitions element structure of alternating groups
general linear group of degree two GL(2,q) over a finite field q (size of field) q^2 - 1 element structure of general linear group of degree two
projective general linear group of degree two PGL(2,q) over a finite field q (size of field) q + 2 if q odd, q + 1 if q a power of 2 element structure of projective general linear group of degree two
special linear group of degree two SL(2,q) over a finite field q (size of field) q + 4 if q odd, q + 1 if q a power of 2 element structure of special linear group of degree two
projective special linear group of degree two PSL(2,q) over a finite field q (size of field) (q + 5)/2 if q odd, q + 1 if q a power of 2 element structure of projective special linear group of degree two
general linear group of degree three GL(3,q) over a finite field q (size of field) q^3 - q = q(q - 1)(q + !) element structure of general linear group of degree three