Isoclinic groups have same proportions of conjugacy class sizes
Suppose and are finite groups that are isoclinic groups. Suppose is a positive integer. Denote by the number of conjugacy classes of of size and denote by the number of conjugacy classes of of size . Then, is nonzero if and only if is nonzero, and if so, we have:
In other words, given the conjugacy class size statistics of , we can obtain the conjugacy class size statistics of by scaling the number of occurrences of each conjugacy class size by a factor of .
In particular, if and also have the same order, then they have precisely the same conjugacy class size statistics.
Relation with Schur covering groups
The idea behind the proof is to show that the size of the conjugacy class of an element depends only on its coset modulo the center, and is completely determined by the information of the commutator map. We use Fact (1).
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Given: Two isoclinic groups and , a positive integer . and are respectively the number of conjugacy classes in and of size . Note that the actual number of elements in and with these conjugacy class sizes are and respectively.
To prove: is nonzero if and only if is nonzero, and if so,
Proof: Let be the group and be the group . Denote by the map obtained from the commutator map in either group (we know both maps are equivalent via the isoclinism). Denote by and the quotient maps modulo the respective centers.
|Step no.||Asssertion/construction||Facts used||Given data used (column to be filled)||Previous steps used||Explanation|
|1||For , the centralizer in of any element in is precisely where .||This follows by definition of centralizer.|
|2||For any , the size of the conjugacy class in of any element in is the index of the subgroup in .||Fact (1)||Step (1)||By Fact (1), the conjugacy class size is , which, by the fourth isomorphism theorem, is .|
|3||The set of elements of with conjugacy class size is where is the set of for which the index of the subgroup in is .||Step (2)|
|4||where is as defined in Step (3). Both sides of the equality represent the number of elements in conjugacy classes of size in .||Step (3)||follows directly, since the set is the union of many cosets, each of size .|
|5||where is as defined in Step (3). Both sides of the equality represent the number of elements in conjugacy classes of size in .||Analogous to reasoning for Steps (1)-(4)||Replace and with and throughout. Note that is defined in the same way in both cases.|
|6||is nonzero if and only if is nonzero, and if so, .||Steps (4), (5)||Direct by taking the quotient of the descriptions.|
|7||is nonzero if and only if is nonzero, and if so, .||Step (6)||We have that since the inner automorphism groups are isomorphic, so rearranging gives . Combine with Step (5) to get the result.|