By Kondratev A.S.

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**Example text**

Which, if any, of the representations will be complex? (3) Show that if we deﬁne η : D4 → GL(C, 2) by ρ(κ) = 01 10 π , ρ(β) = e2ı 0 π 0 e− 2 ı , then η deﬁnes a complex representation of D4 . Find four other inequivalent complex representations of D4 , all of which should be of (complex) dimension 1. (4) Find an inﬁnite set of inequivalent 1-complex dimensional representations of SO(2). Which of these representations become equivalent when viewed as real representations? 1 Averaging over G Suppose that (V, G) is a real representation and G is compact Lie group (or ﬁnite).

Show that ΓX = {exp(tX) | t ∈ R} is an Abelian subgroup of G. Find (up to isomorphism) ΓX in case G = SO(3). Show that for ‘most’ X ∈ so(4), the closure of ΓX is isomorphic to T2 . Find the corresponding results for SO(2n) and SO(2n + 1). (These examples are special cases of the fundamental theorem that every compact connected Lie group has a maximal torus Tm and that the set of conjugates gTm g −1 ﬁlls out G. ) (3) Suppose that Γ is a discrete subgroup of Rn . 1 by showing that there exists a linearly independent set g1 , .

Therefore z = ρ(g −1 z, g) ∈ ρ(Z). 1 If G is not compact, then Gx will generally not be closed. Examples are easily constructed using R- or Z-actions. 1 The orbit space for the action of G on X is the quotient topological space X/G. 2 Suppose that X is G-space and G is a compact topological group. Let q : X → X/G denote the orbit map. Then (1) q is an open, closed and proper mapping (inverse images of compact sets are compact). (2) X/G is Hausdorﬀ. Proof. Let U ⊂ X be open. Then p(U ) is open if and only if p−1 (p(U )) is open (quotient topology).

### 2-Local subgroups of finite groups by Kondratev A.S.

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