Number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one
From Groupprops
Statement
Suppose is a prime number and
is a group of prime power order with underlying prime
. Then the number of conjugacy classes of
(which is the same as the number of irreducible representations) is congruent to the order of
modulo
.
Facts used
- Number of irreducible representations equals number of conjugacy classes
- Degree of irreducible representation divides order of group: For a
-group, in particular, this means that it is a power of
.
- Sum of squares of degrees of irreducible representations equals order of group
Proof
Given: A prime number . A finite
-group
of order
with
conjugacy classes.
To prove:
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() |
Fact (1) | ![]() |
||
2 | All irreducible representations of ![]() ![]() ![]() ![]() |
Fact (2) | ![]() ![]() |
[SHOW MORE] | |
3 | Let ![]() ![]() ![]() ![]() |
Steps (1), (2) | Step-combination direct. | ||
4 | ![]() |
Fact (3) | ![]() ![]() |
Step (3) | Step-fact direct. |
5 | ![]() |
[SHOW MORE] | |||
6 | ![]() |
Step (5) | Sum up Step (5) for ![]() | ||
7 | ![]() |
Steps (3), (4), (6) | Step-combination direct. |