Abstract
Everywhere in this section, we write by default G for an arbitrary finite group and \(\mathbb{k}\) for an algebraically closed field such that \(\mathop{\mathrm{char}}\nolimits (\mathbb{k}) \nmid \vert G\vert \).
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Notes
- 1.
Note that this fails if \(\mathop{\mathrm{char}}\nolimits \mathbb{k}\mid \vert G\vert \).
- 2.
See Theorem 5.5 on p. 117.
- 3.
See Proposition 6.1 on p. 133.
- 4.
Do not confuse these additive characters of arbitrary groups with the multiplicative characters of abelian groups considered in Sect. 5.4.2 on p. 111.
- 5.
See Sect. 5.4 on p. 109.
- 6.
See Definition 5.1 on p. 109.
- 7.
- 8.
See Sects. 10.3.1 and 16.1.1 of Algebra I.
- 9.
See formula (6.4) on p. 132.
- 10.
See Examples 18.3, 18.4 of Algebra I.
- 11.
See Examples 10.1, 10.2 of Algebra I.
- 12.
See formula (6.23) on p. 137.
- 13.
In Theorem 10.2 on p. 233 we will see that for every normal abelian subgroup \(H\lhd G\), the dimension of every irreducible G-module divides the index [G: H].
- 14.
- 15.
See Sect. 5.4 on p. 109.
- 16.
- 17.
See the comments to Exercise 6.7.
- 18.
- 19.
See Sect. 6.2.3 on p. 140.
- 20.
Recall that \(Q_{8} =\{ \pm \mathbf{1},\pm \boldsymbol{i},\pm \boldsymbol{j},\pm \boldsymbol{k}\} \subset \mathbb{H}\) is the group of quaternionic units, and D 4 the group of the square.
- 21.
Called the Heisenberg group over \(\mathbb{F}_{2}\).
- 22.
That is, with trivial kernel kerϱ = e.
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Gorodentsev, A.L. (2017). Representations of Finite Groups in Greater Detail. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_6
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