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Group Theory

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Explanations 1

David Speyer's notes on Jordan-Holder for finite groups: PDF Idea is, look at a normal lattice GGG containing large normal subgroups N,N′N,N'N,N′ containing N∩N′N\cap N'N∩N′ (in…
By Victor Wang

Problems 7

Let AAA be a group, and let BBB be a torsion-free group. What can you say about the set Hom⁡(A,B)\operatorname{Hom}(A,B)Hom(A,B)?
If GGG is a finite solvable group, what else can you say about GGG? Mark all that apply.
Which of the following abelian groups have commutative endomorphism rings?
What choices below represent semidirect products of N,HN,HN,H in which NNN must be a normal subgroup?
Let GGG be a group and let G2G^2G2 denote the subset of squares of GGG. What can you say about G2G^2G2?
When does the inclusion map A→A′A\to A'A→A′ induce a map A/B→A′/B′A/B \to A'/B'A/B→A′/B′? Give the most general correct answer.
Which of the following objects is NOT universal (in the sense of universal properties) with respect to the given property?