MAT301 Assignment 3
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and mark pages for each submitted question. You will lose points otherwise.

(1) Does there exist an element of order 42 in $A_{13} ?$ Justify your answer
(2) Let $G$ be a group, $a, b \in G$ and $|a|=20,|b|=18$ and $\langle a\rangle \cap\langle b\rangle \neq{e}$. Prove that $a^{10}=b^{9}$.
(3) Let $H \subset(\mathbb{Q},+)$ be a subgroup generated by finitely many elements. Prove that $H$ is cyclic.
(4) Does there exist an element $\sigma \in S_{15}$ such that $\sigma^{4}=(3256) ?$ Justify your answer.
(5) Let
$$\sigma=\left[\begin{array}{cccccccccccccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \ 12 & 5 & 6 & 15 & 1 & 10 & 20 & 2 & 13 & 14 & 17 & 19 & 18 & 3 & 9 & 4 & 7 & 11 & 8 & 16 \end{array}\right]$$
(a) Find $|\sigma|$;
(b) Is $\sigma$ even or odd?
(6) Let $\alpha=(124)(3521)(542)$
Find $\alpha^{99}$.
(7) Let $\alpha=(284975)(11063)$ and $\beta=(247)(589)(13610)$ be elements in $S_{10}$.
Find $|\langle\alpha\rangle \cap\langle\beta\rangle|$. Justify your answer.
(8) Let $H=\left{\sigma \in A_{7} \mid \sigma^{2}=\varepsilon\right} .$ Is $H$ a subgroup of $A_{7} ?$

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Fall 2018

This is the webpage for MAT301 during Fall 2018. All the course documents will be posted here. We will be using Quercus for the purposes of announcements and recording grades.

This is a course on group theory for math major students (non-specialists). Please click here for the course syllabus (the document includes all the logistic information about the course, in particular the grading scheme and policies regarding missed term work).

Instructor: Payman Eskandari

Office location: 215 Huron St., Room 1012 (located on the 10th floor). Please note that the elevators in the building only go up to the 9th floor. From there you have to take the stairs.

Email address: [email protected]

Office hours: Fridays 12:30-2:30 and Tuesdays 2:30-4:30 in HU1012

TAs: Thaddeus Janisse ([email protected]), Lennart Doppenschmitt ([email protected]), Jack Ding ([email protected])

TA office hours: Mondays 11-12 (Jack) and Wednesdays 1-2 (Lennart) in PG101

Recommended textbook: Contemporary Algebra by Gallian, 9th edition

Lecture notes

Please click here for the last version of the notes. Every week this file will be updated to include the material covered during that week. If you plan to print the notes, keep in mind that the notes are going to “evolve”: the first 10 pages of this week’s version may not be identical to the first 10 pages of next week’s. This is because for instance, new subsections will be added to the Preliminaries section, as needed. (Final version uploaded on Dec 7, 2018.)

If you want to read ahead, here are the notes from a past (Winter 2017) offering of the course.

Assignments

Assignment 1 deadline extended to Friday Sep 21 at the beginning of the lecture (Note: New version uploaded on Sep 14. There was a typo in question 1b, which is now corrected.) Solutions

Assignment 2 submission deadline Friday Oct 5 at 11:59 pm. The solutions are to be submitted on Crowdmark. Solutions

Assignment 3 submission deadline Friday Oct 26 at 11:59 pm. The solutions are to be submitted on Crowdmark. Solutions

Assignment 4 submission deadline Friday Nov 9 at 11:59 pm. The solutions are to be submitted on Crowdmark. Solutions

Assignment 5 submission deadline Monday Nov 26 at 11:59 pm. The solutions are to be submitted on Crowdmark. Solutions

Assignment 6 submission deadline Wednesday Dec 5 at 11:59 pm. The solutions are to be submitted on Crowdmark. Solutions

Other documents

Week 1 tutorial activity sheet

more practice problems for Test 1

Test 1 Solutions

Test 2 Solutions with indications on the marking scheme