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= Assignment #1: Group Theory I, 6.10.13 =
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= Student solutions for Assignment #3 =
[[Media:MA598A_PS_1.pdf| pdf file ]]<br>
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[[Solution sample|Solution Sample]]  
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.
 
*Post link to solution/discussion page here
 
*post link to other solution/discussion page here.
 
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==(2)==
 
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 ).
 
*Post link to solution/discussion page here
 
*post link to other solution/discussion page here.
 
 
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==(3)==
 
(a) Define what it means for a group to be simple.
 
  
(b) True or false: an abelian group is simple if and only if it is cyclic.
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== Problem 50  ==
  
(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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*[[Media:Prob50.pdf|Problem 50 - Tan Dang]]
  
(d) Show that any group of order p2q, p, q prime, is not simple.
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== Problem 73  ==
*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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Show that if <span class="texhtml">''p''</span> is a prime such that there is an integer <span class="texhtml">''b''</span> with <span class="texhtml">''p'' = ''b''<sup>2</sup> + 4</span>, then <math>\mathbb{Z}[\sqrt{p}]</math> is not a unique factorization domain.  
  
(b) Show that exactly one of these subgroups is cyclic.
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*[[Media:Problem_73_Zeller.pdf|Solution by Andrew Zeller]]
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::Here's my alternate proof - I found a few things that need to be changed in Avi's, which I can discuss in class.
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*[[Media:Prob_73.pdf|Solution by Avi Steiner]]
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::My solution only uses the fact that ''p'' is a sum of two squares (i.e. is congruent to 1 mod 4), so I'm not sure it's correct. -- Avi 20:05, 25 June 2013 (UTC)
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*[[MA553QualStudyAssignment3Problem73Solution|Solution by Ryan Spitler]]
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::I think this is a bit cleaner.
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::: I very much prefer this solution! -- Avi 11:08, 26 June 2013 (UTC)
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== Problem 94  ==
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Show <span class="texhtml">''f''(''x'') = ''x''<sup>4</sup> + 5''x''<sup>2</sup> + 3''x'' + 2</span> is irreducible over the field of rational numbers.
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*[[Media:Problem_94_-_Nicole_Rutt.pdf|Solution by Nicole_Rutt]]
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== Problem 101  ==
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(a) Show that <span class="texhtml">''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1</span> is irreducible in <math>\mathbb{Z}_3[x]</math>.
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(b) Show that <span class="texhtml">''x''<sup>4</sup> + 1</span> is not irreducible in <math>\mathbb{Z}_3[x]</math>
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*[[Media:Week_3_Problem_101.pdf|Solution]]
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== Problem 107  ==
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Let <span class="texhtml">''R''</span> be a commutative ring with identity such that the identity map is the only ring automorphism of <span class="texhtml">''R''</span>. Prove that the set <span class="texhtml">''N''</span> of all nilpotent elements of <span class="texhtml">''R''</span> is an ideal of <span class="texhtml">''R''</span>
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*[[Assn3Prob107|Solution by Avi Steiner]]
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== Problem 114  ==
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A local ring is a commutative ring with 1 that has a unique maximal ideal. Show that a ring R is local if and only if the set of non-units in R is an ideal.
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*[https://kiwi.ecn.purdue.edu/rhea/images/c/ca/Problem_114_Kent_State_Algebra_Qual_Ring.pdf Solution by Nathan Moses]<br>
  
(c) Show that the intersection of these subgroups is the commutator subgroup of D4.
 
*Post link to solution/discussion page here
 
*post link to other solution/discussion page here.
 
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==(6)==
 
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?
 
*Post link to solution/discussion page here
 
*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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[[Category:MA598ASummer2013Weigel]] [[Category:Math]] [[Category:MA598]] [[Category:Problem_solving]] [[Category:Algebra]]
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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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----
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[[2013 Summer MA 598A Weigel|Back to 2013 Summer MA 598A Weigel]]
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Latest revision as of 06:11, 26 June 2013


Student solutions for Assignment #3

Solution Sample


Problem 50

Problem 73

Show that if p is a prime such that there is an integer b with p = b2 + 4, then $ \mathbb{Z}[\sqrt{p}] $ is not a unique factorization domain.

Here's my alternate proof - I found a few things that need to be changed in Avi's, which I can discuss in class.
My solution only uses the fact that p is a sum of two squares (i.e. is congruent to 1 mod 4), so I'm not sure it's correct. -- Avi 20:05, 25 June 2013 (UTC)
I think this is a bit cleaner.
I very much prefer this solution! -- Avi 11:08, 26 June 2013 (UTC)

Problem 94

Show f(x) = x4 + 5x2 + 3x + 2 is irreducible over the field of rational numbers.

Problem 101

(a) Show that x4 + x3 + x2 + x + 1 is irreducible in $ \mathbb{Z}_3[x] $.

(b) Show that x4 + 1 is not irreducible in $ \mathbb{Z}_3[x] $

Problem 107

Let R be a commutative ring with identity such that the identity map is the only ring automorphism of R. Prove that the set N of all nilpotent elements of R is an ideal of R

Problem 114

A local ring is a commutative ring with 1 that has a unique maximal ideal. Show that a ring R is local if and only if the set of non-units in R is an ideal.


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