14-1100/Homework Assignment 2
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The information below is preliminary and cannot be trusted! (v)
This assignment is due at class time on Thursday, October 20, 2010.
Solve the following questions
- (Selick)
- What it the least integer [math]\displaystyle{ n }[/math] for which the symmetric group [math]\displaystyle{ S_n }[/math] contains an element of order 18?
- What is the maximal order of an element in [math]\displaystyle{ S_{26} }[/math]? (That is, of a shuffling of the red cards within a deck of cards?)
- (Selick) Let [math]\displaystyle{ H }[/math] be a subgroup of index 2 in a group [math]\displaystyle{ G }[/math]. Show that [math]\displaystyle{ H }[/math] is normal in [math]\displaystyle{ G }[/math].
- Let [math]\displaystyle{ \sigma\in S_{20} }[/math] be a permutation whose cycle decomposition consists of one 5-cycle, two 3-cycles, and one 2-cycle. What is the order of the centralizer [math]\displaystyle{ C_{S_{20}}(\sigma) }[/math] of [math]\displaystyle{ \sigma }[/math]?
- (Selick) Let [math]\displaystyle{ G }[/math] be a group of odd order. Show that [math]\displaystyle{ x }[/math] is not conjugate to [math]\displaystyle{ x^{-1} }[/math] unless [math]\displaystyle{ x=e }[/math].
- (Dummit and Foote) Show that if [math]\displaystyle{ G/Z(G) }[/math] is cyclic then [math]\displaystyle{ G }[/math] is Abelian.
- (Lang) Prove that if the group of automorphisms of a group [math]\displaystyle{ G }[/math] is cyclic, then [math]\displaystyle{ G }[/math] is Abelian.
- (Lang)
- Let [math]\displaystyle{ G }[/math] be a group and let [math]\displaystyle{ H }[/math] be a subgroup of finite index. Prove that there is a normal subgroup [math]\displaystyle{ N }[/math] of [math]\displaystyle{ G }[/math], contained in [math]\displaystyle{ H }[/math], so that [math]\displaystyle{ (G:N) }[/math] is also finite. (Hint: Let [math]\displaystyle{ (G:H)=n }[/math] and find a morphism [math]\displaystyle{ G\to S_n }[/math] whose kernel is contained in [math]\displaystyle{ H }[/math].)
- Let [math]\displaystyle{ G }[/math] be a group and [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ H_2 }[/math] be subgroups of [math]\displaystyle{ G }[/math]. Suppose [math]\displaystyle{ (G:H_1)\lt \infty }[/math] and [math]\displaystyle{ (G:H_2)\lt \infty }[/math]. Show that [math]\displaystyle{ (G:H_1\cap H_2)\lt \infty }[/math]
Problem 1. (Selick) Show that any group of order 56 has a normal Sylow-[math]\displaystyle{ p }[/math] subgroup, for some prime [math]\displaystyle{ p }[/math] dividing 56.
Problem 2. (Qualifying exam, May 1997) Let [math]\displaystyle{ S_5 }[/math] act on [math]\displaystyle{ ({\mathbb Z/5})^5 }[/math] by permuting the factors, and let [math]\displaystyle{ G }[/math] be the semi-direct product of [math]\displaystyle{ S_5 }[/math] and [math]\displaystyle{ ({\mathbb Z/5})^5 }[/math].
- What is the order of [math]\displaystyle{ G }[/math]?
- How many Sylow-5 subgroups does [math]\displaystyle{ G }[/math] have? Write down one of them.
Problem 3. (Selick) Show that the group [math]\displaystyle{ Q }[/math] of unit quaternions ([math]\displaystyle{ \{\pm 1, \pm i, \pm j, \pm k\} }[/math], subject to [math]\displaystyle{ i^2=j^2=k^2=-1\in Z(Q) }[/math] and [math]\displaystyle{ ij=k }[/math]) is not a semi-direct product of two of its proper subgroups.
Problem 4. (Qualifying exam, September 2008) Let [math]\displaystyle{ G }[/math] be a finite group and [math]\displaystyle{ p }[/math] be a prime. Show that if [math]\displaystyle{ H }[/math] is a [math]\displaystyle{ p }[/math]-subgroup of [math]\displaystyle{ G }[/math], then [math]\displaystyle{ (N_G(H):H) }[/math] is congruent to [math]\displaystyle{ (G:H) }[/math] mod [math]\displaystyle{ p }[/math]. You may wish to study the action of [math]\displaystyle{ H }[/math] on [math]\displaystyle{ G/H }[/math] by multiplication on the left.
