# Linear representation theory of groups of prime-cube order

## Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of prime-cube order.
View linear representation theory of group families | View other specific information about groups of prime-cube order

The discussion here is specifically intended to cater to odd primes $p$ and excludes the case $p = 2$, which is somewhat different. For more on $p = 2$, see linear representation theory of groups of order 8.

## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

Due to the fact that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, for groups of order $p^3$ for a fixed prime $p$, the nilpotency class determines the degrees of irreducible representations. (This also holds for $p = 2$, although we are not discussing that case.

Group GAP ID second part Degrees as list Number of irreps of degree 1 (= order of abelianization) Number of irreps of degree $p$ Total number of irreps (= number of conjugacy classes)
cyclic group of prime-cube order 1 1 ( $p^3$ times) $p^3$ 0 $p^3$
direct product of cyclic group of prime-square order and cyclic group of prime order 2 1 ( $p^3$ times) $p^3$ 0 $p^3$
unitriangular matrix group:UT(3,p) 3 1 ( $p^2$ times), $p$ ( $p - 1$ times) $p^2$ $p - 1$ $p^2 + p - 1$
semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 1 ( $p^2$ times), $p$ ( $p - 1$ times) $p^2$ $p - 1$ $p^2 + p - 1$
elementary abelian group of prime-cube order 5 1 ( $p^3$ times) $p^3$ 0 $p^3$

## Splitting field

Two important notes:

With these facts in mind, we present the list of minimal splitting fields. We denote by $\zeta_n$ a primitive $n^{th}$ root of unity. Note that the degree of the extension $\mathbb{Q}(\zeta_n)$ over $\mathbb{Q}$ is the Euler totient function $\varphi(n)$.

Group GAP ID second part Nilpotency class minimal splitting field = field generated by character values (because it's an odd-order p-group) degree of extension over $\mathbb{Q}$ minimal sufficiently large field degree of extension over $\mathbb{Q}$ Note
cyclic group of prime-cube order 1 1 $\mathbb{Q}(\zeta_{p^3})$ $p^2(p - 1)$ $\mathbb{Q}(\zeta_{p^3})$ $p^2(p - 1)$ both same, because abelian
direct product of cyclic group of prime-square order and cyclic group of prime order 2 1 $\mathbb{Q}(\zeta_{p^2})$ $p(p - 1)$ $\mathbb{Q}(\zeta_{p^2})$ $p(p - 1)$ both same, because abelian
unitriangular matrix group:UT(3,p) 3 2 $\mathbb{Q}(\zeta_p)$ $p - 1$ $\mathbb{Q}(\zeta_p)$ $p - 1$
semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 2 $\mathbb{Q}(\zeta_p)$ $p - 1$ $\mathbb{Q}(\zeta_{p^2})$ $p(p - 1)$ splitting not implies sufficiently large
elementary abelian group of prime-cube order 5 1 $\mathbb{Q}(\zeta_p)$ $p - 1$ $\mathbb{Q}(\zeta_p)$ $p - 1$ all same, because abelian