Abstract Nonsense

Crushing one theorem at a time

University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) ( January 2003))


Point of Post: A friend of mine who attended UMD CP for his Ph.D. has told me that they are famous for their brutally hard qualifying exams. Thus, to keep sharp and as a challenge I’d like to complete all of the qual exams for Algebra, Analysis, and Topology/Geometry eventually–although this is an ambitious venture. So, I plan to break them up into subsections and do all of the ____ subject (ring, field,…) in the Algebra quals and then maybe do all the ____  subject( measure theory, complex analysis, …) in the Analysis quals etc. Since what I’m currently doing on my blog has been representation theory and group theory mostly I thought I’d start with those two. Thus, this will be the first post in a series to solve the Group Theory/Representation Theory problems on the quals.

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NOTE: Just because I am posting references (which are really more like further reading) I will not use any book while taking these.

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NOTE: Instead of doing this post by post I thought it would make more sense to have it all consolidated. See here for the up-to-date PDF of the parts of the exam I’ve finished.

Remark: The set of all the algebra exams may be found here.

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Algebra Qual(Ph.D. Version)-January 2003

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Problem 1:

a) Let H be a group of order 9. Show that \left|\text{Aut}(H)\right|\mid 48 (Hint: you may assume that the group \text{GL}_2\left(\mathbb{F}_3\right) has order 48).

b) Let G be a group of order 153=3^2\cdot17. Show that \mathcal{Z}(G) contains a subgroup of order 9.

c) Find all groups of order 153.

(Do not use theorems of the form “every group of order p^2q…”)

Proof: 

a) Since |H|=3^2 we know that H\cong\mathbb{Z}_9 or H\cong\mathbb{Z}_3^2. Thus, it assumes to prove this is true for those two groups. So, first define F:\text{Aut}\left(\mathbb{Z}_3^2\right)\to\text{Mat}_2\left(\mathbb{Z}_3\right) by

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\displaystyle F\left(\varphi\right)=\begin{pmatrix}\pi_1(\varphi((1,0)) & \pi_1(\varphi(0,1))\\ \pi_2\left(\varphi(1,0)\right) & \pi_2\left(\varphi((0,1))\right)\end{pmatrix}

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\pi_i is the i^{\text{th}} canonical projection. Evidently \mathbb{Z}_3^2 is a vector space over \mathbb{Z}_3 and evidently every group endomorphism will be a linear endomorphism. Thus, since by definition F(\varphi) is by definition \left[\varphi\right]_{\mathcal{B}} where \mathcal{B} is the ordered basis \left((1,0),(0,1)\right) we have from basic linear algebra that \text{im}(F)\subseteq\text{GL}\left(\mathbb{F}_3,2\right) and F is a monomorphism. Thus F\cong\text{im}(F)\leqslant \text{GL}_2\left(\mathbb{F}_3\right) and so by Lagrange’s theorem \left|\text{Aut}\left(\mathbb{Z}_3^2\right)\right|\mid \left|\text{GL}\left(\mathbb{F}_3\right)\right|=48.

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For \mathbb{Z}_9 we appeal to the common fact that \text{Aut}\left(\mathbb{Z}_n\right)\cong\mathbb{Z}_n^\times and so \left|\text{Aut}\left(\mathbb{Z}_9\right)\right|=\varphi\left(9\right)=6\mid 48 where here \varphi is Euler’s totient function.

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b) Let n_3=\#\left(\text{Syl}_3\left(G\right)\right) denote the number of Sylow 3-subgroups of G. By Sylow’s theorems we must have that n_3\mid 153 and so n_3=1,3,9,19,17\cdot3,153 but since n_3\equiv1\text{ mod }3 we must conclude that n_3=1. Thus, by a common theorem we may conclude that if H is the unique Sylow 3-subgroup that H\unlhd G. Thus, consider the map \Phi:G\to\text{Aut}(H) by g\mapsto i_g where i_g is the inner automorphism (this is well defined since H is normal). This is evidently a homomorphism and so what we have by the first isomorphism theorem is that G/\ker\Phi\cong\text{im}(\Phi)\leqslant \text{Aut}(H) and so in particular \left|G/\ker\Phi\right|\mid \text{Aut}(H). But, by definition \ker\Phi=\bold{C}_G(H) the centralizer of H in G and so by part a) we may conclude that \left|G/\bold{C}_G(H)\right|\mid 48. But, since \left(48,153\right)=1 we may conclude that G/\bold{C}_G(H) is trivial and so \bold{C}_G(H)=G. Thus, H\leqslant \mathcal{Z}(G).

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c) We use the fact that since \left|\mathcal{Z}\left(G\right)\right|=9,153 that G/\mathcal{Z}(G) is inevitably cyclic and so by a common theorem we must have that G is abelian. Then, by the structure theorem we may conclude that either G\cong\mathbb{Z}_9\times\mathbb{Z}_{17} or G\cong\mathbb{Z}_3\times\mathbb{Z}_3\times\mathbb{Z}_{17}.

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References:

1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

2. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

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May 1, 2011 - Posted by | Algebra, Fun Problems, Group Theory, Representation Theory, UMaryland Qualifying Exams | , , , , ,

3 Comments »

  1. […] Point of Post: This is the August 2003 part of the post started here. […]

    Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August-2003) « Abstract Nonsense | May 1, 2011 | Reply

  2. […] Point of Post: This is the August 2003 part of the post started here. […]

    Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (January-2004) (Pt. I) « Abstract Nonsense | May 6, 2011 | Reply

  3. […] Point of Post: This is the August 2004 part of the post started here. […]

    Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August-2004) « Abstract Nonsense | May 6, 2011 | Reply


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