Topological Groups 2. Exercise Sheet
Department of Mathematics Summer Term 2013
Andreas Mars 8.5.2013
Groupwork
Exercise G1 (Identity Neighbourhoods) LetGbe a topological group. Show:
(a) For any identity-neighbourhoodU∈Uthere exists an (open) identity-neighbourhoodVwith the properties:
V V⊆U, V−1V⊆U, V V−1⊆U, V−1V−1⊆U.
(b) The multiplication mapµG:G×G→Gis open.
(c) LetY ⊆Gbe a connected subset. ThenY is connected as well. (NB: This is true for all topological spaces.) Exercise G2 (Normal Subgroups)
LetGbe an abstract group and letNÃGbe a normal subgroup. Assume thatN is a topological group and the action of GonN by conjugation is continuous. Show that there exists a unique group topology onGsuch that
(a) the induced topology onN coincides with the original topology and (b) N is open inG.
You might want to prove the following Lemma along the way: A subgroupH≤Gis open if and only if it contains an identity neighbourhood.
Exercise G3 (Closed Sets)
Let f:G→Hbe a morphism and letHbe Hausdorff. Then the graphX :={(x,y)∈G×H| f(x) = y}of f is closed inG×H.
Hence or otherwise, show that for any family of Hausdorff groupsXnand fn:Xn+1→Xnmorphisms the set{(xn)| (∀n∈N): fn(xn+1) =xn}is a closed subset ofQ
Xn. Exercise G4 (The Neighbourhood filter of the identity)
(a) Find an example of a topological groupG and two closed subsetsA,B ⊆G such thatAB ⊆G isnotclosed (cf.
Proposition 2.12 (ii)).
(b) Prove Corollary 2.16: LetGbe a topological group andHbe Hausdorff. Then for each morphism f:G→Hthere is a unique morphism f0:G/{1} →Hsuch thatf =f0◦q, whereq:G→G/{1}is the quotient morphism.
Homework
Exercise H1 (Decomposition of Morphisms)
Prove Proposition 2.10 (canonical decomposition of morphisms).
Exercise H2 (A Clopen Intersection) Prove Lemma 2.19.
Exercise H3 (Characteristic Subgroups) Prove or give a counterexample:
(a) LetN ÃGbe a normal subgroup ofG. ThenN is characteristic.
(b) IfGis abelian (in particular, every subgroup is normal), then every subgroup is characteristic.
Exercise H4 (Construction of Topologies)
In case Theorem 2.22 does not seem familiar to you, prove it.
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