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[[Category:Algebra]]   
  
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= Assignment #1: Group Theory I, 6.10.13  =
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= Assignment #3 =
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[[Media:MA598A_PS_1.pdf| pdf file ]]<br>
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Please post comments, questions, attempted or completed solutions, etc. here.&nbsp; If you want to post a solution, create a new page using the toolbar on the left.
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==(1)==
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(a) Define normal subgroup. Be as succinct as possible.
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(b) Let G be a finite group with |G|=n, and suppose H is a subgroup of G with |G:H| = p, with p the smallest prime divisor of n. Show that H is normal in G.
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==(50)==
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[[Media:Prob50.pdf‎|Problem 50]]
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'''Comments/Answers'''
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*For a), would we need to define everything, starting from a group, and then subgroup?
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(2)==
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Suppose that (V, < , >) is an inner product space. True or false: the set of isometries of V (i.e. the set of automorphisms preserving the inner product) is a proper subgroup of Aut(V ).
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(3)==
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(a) Define what it means for a group to be simple.
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(b) True or false: an abelian group is simple if and only if it is cyclic.
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(c) Let Sn denote the symmetric group on n letters. • An, the set of odd permutations in Sn, is a normal subgroup. • Assume that A5 is simple. Show that An is simple for all n ≥ 5.
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(d) Show that any group of order p2q, p, q prime, is not simple.
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(4)==
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Show that any group G for which Aut(G) is cyclic must be abelian.
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(5)==
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Let D4 = ⟨(1234), (12)(34)⟩ ⊂ S4 be the Dihedral group of order 8.
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(a) Show that D4 has exactly three subgroups of order 4.
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(b) Show that exactly one of these subgroups is cyclic.
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(c) Show that the intersection of these subgroups is the commutator subgroup of D4.
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(6)==
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Let G be a finite group and H a proper subgroup of G. Show that there exists an element of G which is not conjugate to any element of H. Does this remain true if G is allowed to be infinite?
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(7)==
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Let G be a group of odd order. Show that if g ∈ G−{e}, then g is not conjugate to its inverse.
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(8)==
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Determine the number of pairwise non-isomorphic groups of order 15.
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(9)==
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Suppose that G is a group and φ ∈ Hom(G), g 􏰭→ g2, • Show that there exists a finite nonabelian group where φ(g) = g4 is a homomorphism.
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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==(10)==
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Suppose φ:G×G→G is a homomorphism and that there exists n ∈ G such that φ(n,g)=φ(g,n)=g, for all g∈G. Show that G is abelian and φ is simply group multiplication.
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*Post link to solution/discussion page here
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*post link to other solution/discussion page here.
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[[2013 Summer MA 598A Weigel|Back to 2013 Summer MA 598A Weigel]]
 
[[2013 Summer MA 598A Weigel|Back to 2013 Summer MA 598A Weigel]]

Revision as of 18:52, 23 June 2013


Assignment #3

(50)

Problem 50

Back to 2013 Summer MA 598A Weigel

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