# Number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one

## 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 | equals the number of irreducible representations of . | Fact (1) | is the number of conjugacy classes. | ||

2 | All irreducible representations of have degree for some nonnegative integer , with . | Fact (2) | has order | [SHOW MORE] | |

3 | Let be the number of irreducible representations of degree for . Then, . | Steps (1), (2) | Step-combination direct. | ||

4 | Fact (3) | has order | Step (3) | Step-fact direct. | |

5 | [SHOW MORE] | ||||

6 | Step (5) | Sum up Step (5) for . | |||

7 | Steps (3), (4), (6) | Step-combination direct. |