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Section 13: 1, 2, 4, 5, 8, 18, 20, 23, 36, 43, 50, 52
Section 14: 1, 5, 17,; 31
Section 15: 37 (Extra Credit)

Section 10: 3, 4, 6, 15, 19abcde, 20, 21, 22, 23, 28, 29

Wiki:
2. Write out some (or all) of the cosets of your subgroup from 1.

Section 8: 1, 2, 5, 6, 9, 11, 12, 16, 30, 33, 35cefgh, 40, 46, 48.

Extra Problem:

Let $G$ be a group, and let $H$ be a subgroup. We define a relation $\sim_H$ on the group in the following way. For any two elements $a,b\in G$, we let $a\sim_H b$ if and only if $ab^{-1}\in H$. Prove that $\sim_H$ is an equivalence relation.

Section 5: 2, 3, 6, 9, 21, 41, 43, 52, 54
Section 6: 1, 4, 7, 12, 17, 22, 32ab, 33, 46, 52, 55

Josh,

Yes, it is the number of even permutations inside of S5. The only question remaining is what even permutation of S5 means formally. The book may define it, but I am not sure what it says. This group may require you to understand S5 first, and then understand A5 as a subgroup.

Re: Adopt A Group Project by , 01 Oct 2009 05:29

So just to get a little bit clearer of a picture (but yes will be coming to see you tomorrow afternoon) the order of the alternating group on the set {1,2,3,4,5} is the number of even permutations on A5? Thanks again. See you tomorrow.

Re: Adopt A Group Project by , 01 Oct 2009 04:46

Josh,

Come by sometime to talk about this. Essentially, a permutation is a function mapping the set {1,2,3,4,5} 1-1 and onto itself. Any function will do, and is a permutation. An alternating permutation is one that "swaps an even number of things." For instance the function that swaps 1 and 2 and fixes 3, 4, and 5 is not an even permutation, but the function that swaps 1 and 2; and 3 and 4; and fixes 5, will be an even permutation. This may take a while before we get to it in class, so you might be better off coming to discuss it with me.

Re: Adopt A Group Project by , 01 Oct 2009 04:33

Dr. Edgar,

So what exactly is a permutation group. I am trying to figure out what an alternating group is but I have yet to even figure out what my operation is? I know what my set is but I don't know my operation and I think it has to do something with the fact that everywhere I look it says it is a permutation group. Is the permutations the number of times a number can be "alternated" with another number in the set? ie. swapped? Also, is there a set number of permutations allowed in the A5 set? Sorry for all the questions but this is burning me up right now. Thanks.

Josh

Adopt A Group Project by , 01 Oct 2009 03:55

Section 3: 2, 3, 4, 8, 21, 26, 30
Section 4: 41

Section 3: 11, 12, 17, 27

and Prove: The inverse property is a structural property.

in other words, Let $(S,*)\cong (T,\star)$ be isomorphic binary structures via the bijection $\varphi:S\to T$. If $a\in S$ has an inverse, $a'$, then $\varphi(a)$ also has an inverse and $\varphi(a)'=\varphi(a')$ as expected.

Section 1: 22, 23, 31, 33, 34
Section 2: 1, 2, 4, 7, 8, 17, 23, 24adgj, 26, 27, 29, 36.

Section 4: Pick four from (1, 2, 3, 5, 6, 7, 8); Do 10a, 12, 19, 29, 32, (33 or 35), 36.

The homework due on Friday is as below. Please check this regularly as I will add to it after Monday, September 14.

Section 0: 1, 3, 4, 6, 8, 9, 12af, 14a, 15, 25, 30, 31, 34, 35c, 36a.

Bonus Problem: Section 0 #18