Character table-equivalent groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Suppose $G_{1}$ and $G_{2}$ are finite groups and $K$ is a splitting field in characteristic zero for both (we can take $K=\mathbb {C}$ or ${\overline {\mathbb {Q} }}$. We say that $G_{1}$ and $G_{2}$ are character table-equivalent groups if there exists a bijection $\alpha$ between the set of conjugacy classes in $G_{1}$ and that of $G_{2}$, and a bijection $\beta$ between the set of irreducible characters (up to equivalence) of $G_{1}$ and that of $G_{2}$, such that under this pair of bijections, the character table of $G_{1}$ is identified with that of $G_{2}$. In other words, for any character $\chi$ of $G_{1}$ and any conjugacy class $c$ of $G_{1}$, we have $\beta (\chi )(\alpha (c))=\chi (c)$.

Invariants

Character table-equivalent groups share a number of properties and arithmetic function values.

Invariant	Why it's the same for character table-equivalent groups
degrees of irreducible representations	These can be read off from the character table as the set of values that the character table takes at the identity element's conjugacy class. Note that the identity element's conjugacy class is the unique conjugacy class at which all character values are positive (this can be deduced from the column orthogonality theorem).
order of the whole group	Follows from knowledge of degrees of irreducible representations, because sum of squares of degrees of irreducible representations equals order of group
conjugacy class sizes	The size of each conjugacy class can be computed using the order and by computing its norm and using the column orthogonality theorem.
order of derived subgroup (or equivalently, its index, which is the order of the abelianization)	use number of one-dimensional representations equals order of abelianization
order of center	number of conjugacy classes of size 1.
field generated by character values	The character table entries are the same, so the field they generate is the same.

On the other hand, the following are not determined:

• Minimal splitting field: We could have two character table-equivalent groups such that they do not have isomorphic minimal splitting fields. For instance, dihedral group:D8 and quaternion group (see linear representation theory of dihedral group:D8 and linear representation theory of quaternion group).