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April 26

Understanding the structure of order 16 groups

So I have been attempting to gain a better understanding of the 14 different groups (up to isomorphism) of order 16 and I am having some difficulties finding ways make sense of their similarities and differences. With the exception of the Alternating group on a 4-element set ( $A_{4}$ ), I understand the structure of all groups of orders<16 well enough that I could produce an operation table for them without any references. I am wanting to gain a similar faculty with order 16 groups without resorting to something like rote memorization (which would be devoid of real understanding and kinda defeats the point of learning it for me). What sort of things should I look at beyond what I already have been using? (I have been tackling this problem by so far by using GAP to get an operation table for the group, then I proceed to manually finding the various cycles, identifying which ones are primitive (meaning not a subset of a larger cycle), and then examining how their generators behave when operated together). 76.14.232.104 (talk) 10:06, 26 April 2015 (UTC)[reply]

Are you already familiar with the Sylow theorems? Burnside's_lemma might also help. SemanticMantis (talk) 14:56, 26 April 2015 (UTC)[reply]

It's unclear to me what "understanding the structure" means exactly, but this list looks like a good place to start. There are five abelian groups corresponding to the five partitions of 4. There are three non-abelian split extensions of Z8 by Z2: the dihedral, quasidihedral, and M16 groups. Two groups are gotten from the non-abelian groups of order 8 by taking direct products with Z2. There is a unique non-abelian split extension of Z4 by Z4, a split extension of Z2xZ4 by Z2, and a central product of D8 and Z4. Finally, there is the generalized quaternion group of order 16. Sławomir Biały (talk) 15:20, 26 April 2015 (UTC)[reply]

I've worked out generators and relations for groups of 16 and of order 32. To get to 64 you need more sophisticated techniques than I know about and for 128 you also need a computer. A couple of things I learned though: First, what most textbooks tell you about classifying groups of order 8 (i.e. starting with a large cyclic subgroup) doesn't scale and you need to do something else for order 16. The most important elementary theorem to know is that G/Z can't be cyclic. There is a generalization of this which gives a criterion for which groups are possible G/Z, turns out that not many are. So it's relatively easy to enumerate possible Z's and G/Z's. The next thing is to learn a bit about the machinery of central extensions. This naturally leads to more sophisticated group homology theory but you probably don't need to go that far. Also, learn what you can about Frattini subgroups and their quotients. You will need a bit more linear algebra that they normally teach undergraduates as well, specifically the classification of alternating bilinear forms. As a rule of thumb, every time you multiply the order by 2 the classification problem gets an order of magnitude more difficult. So 16 isn't just a lengthier version of 8, it's more like comparing a symphony with a pop song. (Which makes 32 Wagner's Ring cycle I guess.) --RDBury (talk) 17:44, 26 April 2015 (UTC)[reply]

Sum of Sine (Not Homework)

If a, b, c, d, e, f,......, z are in Geometric Progression, then what would be the sum of following series?
1. (Sin x) + (Sin ax) + (Sin bx) + (Sin cx) + ........ + (Sin zx)
2. (Sin x) * (Sin ax) * (Sin bx) * (Sin cx) * ........ * (Sin zx)
(+ denotes addition and * denotes multiplication)
Please, suggest me, if there is any article related to this. --106.215.138.165 (talk) 20:07, 26 April 2015 (UTC)[reply]

I think both diverge for nonzero x. However the first one is somewhat related to the Weierstrass function and is a formal Fourier series. --Jasper Deng (talk) 20:16, 26 April 2015 (UTC)[reply]