Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order
From Groupprops
This article gives a fact that is true for small groups of prime power order.More specifically, it is true for all groups of orderwhere
is at most equal to 4.
See more such facts| See more facts true for prime powers up to the same maximum power 4
Statement
Suppose is a prime number and
. Then, for a group of order
, the Nilpotency class (?)
of the group determines its conjugacy class size statistics.
Here is the complete list:
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number of conjugacy classes of size 1 | number of conjugacy classes of size ![]() |
number of conjugacy classes of size ![]() |
---|---|---|---|---|
0 | 0 | 1 | 0 | 0 |
1 | 1 | ![]() |
0 | 0 |
2 | 1 | ![]() |
0 | 0 |
3 | 1 | ![]() |
0 | 0 |
3 | 2 | ![]() |
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0 |
4 | 1 | ![]() |
0 | 0 |
4 | 2 | ![]() |
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0 |
4 | 3 | ![]() |
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