Number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one
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 .
- 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
Given: A prime number . A finite -group of order with conjugacy classes.
|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.|
|6||Step (5)||Sum up Step (5) for .|
|7||Steps (3), (4), (6)||Step-combination direct.|