# Linear representation theory of groups of order 2^n

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This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 2^n.

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This article describes the linear representation theory of groups of order 2^n, i.e., groups whose order is a power of .

## 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 subgroupSize 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 subgroupCumulative 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

Number of degree 1 irreps | Number of degree 2 irreps | Number of degree 4 irreps | Number of degree 8 irreps | Total number of irreps | Order of group (also, sum of squares of degrees) | Number of groups with these degrees of irreps | Description of the groups |
---|---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 1 | 1 | 1 | trivial group only |

2 | 0 | 0 | 0 | 2 | 2 | 1 | cyclic group:Z2 only |

4 | 0 | 0 | 0 | 4 | 4 | 2 | Both the groups of order 4: cyclic group:Z4 and Klein four-group |

8 | 0 | 0 | 0 | 8 | 8 | 3 | The abelian groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8 |

4 | 1 | 0 | 0 | 5 | 8 | 2 | The non-abelian groups of order 8: dihedral group:D8 (see representation info) and quaternion group (see representation info) |

16 | 0 | 0 | 0 | 16 | 16 | 5 | The abelian groups of order 16: cyclic group:Z16, direct product of Z4 and Z4, direct product of Z4 and Z4, direct product of Z8 and Z2, direct product of Z4 and V4, elementary abelian group:E16 |

8 | 2 | 0 | 0 | 10 | 16 | 6 | The groups of order 16, class exactly two, the Hall-Senior family : SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4 |

4 | 3 | 0 | 0 | 7 | 16 | 3 | The groups of order 16, class exactly three, the Hall-Senior family : dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 |

32 | 0 | 0 | 0 | 32 | 32 | 7 | All the abelian groups of order 32 |

16 | 4 | 0 | 0 | 20 | 32 | 15 | The Hall-Senior family (up to isocliny) |

16 | 0 | 1 | 0 | 17 | 32 | 2 | The extraspecial groups (Hall-Senior family ): inner holomorph of D8 and central product of D8 and Q8 |

8 | 6 | 0 | 0 | 14 | 32 | 19 | The Hall-Senior families (ten groups, class three) and (nine groups, class two) |

8 | 2 | 1 | 0 | 11 | 32 | 5 | The Hall-Senior families and |

4 | 7 | 0 | 0 | 11 | 32 | 3 | The maximal class groups (family ): dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32 |

## Splitting field

### Important ways in which 2 differs from other primes

There are two very important differences between 2 and other primes:

- Odd-order and ambivalent implies trivial, so a nontrivial finite -group for odd cannot be ambivalent, i.e., not all its characters are real-valued. In particular, this means that the representations cannot all be realized over the field of real numbers, and in particular, there are no nontrivial examples of rational-representation groups or rational groups for odd primes. However, for the prime , there are many examples of rational-representation groups (such as elementary abelian 2-groups and dihedral group:D8), examples of rational groups that are not rational-representation groups (such as quaternion group), and examples of other ambivalent groups (such as dihedral group:D16).
- Odd-order p-group implies every irreducible representation has Schur index one: This means that for an odd-order -group, every irreducible representation can be realized over the field generated by its character values (
*Note*: There do exist non-nilpotent odd-order groups with representations having Schur index values more than 1). This is*not*the case for 2-groups, and there exist irreducible representations of 2-groups with Schur index greater than 1. The smallest example is faithful irreducible representation of quaternion group, which has Schur index 2. - The multiplicative group of is cyclic for odd , but is
*not*cyclic for . Thus, we can construct examples of finite 2-groups such that the Galois group of a minimal cyclotomic splitting field over is not cyclic, and this is not possible for odd . Therefore, 2 is the*only*prime where it is possible to construct examples where the orbit structure on the irreducible representations*differs*from the orbit structure on the conjugacy classes under the action of a Galois group. (Examples, links need to be provided).

### Information on splitting fields

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