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Math Help - Elements generated by...

  1. #1
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    Elements generated by...

    What are the elements of <(15), (243)>? The comma is throwing me off... Is it simply all elements generated by (15) AND (243)? Or the product of the two? Can someone show the elements and how you got them? Thanks
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  2. #2
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    Re: Elements generated by...

    what KIND of structure are you generating? we're not mind-readers. and what is (15)? notations vary, do you mean the ideal of integers generated by 15, or the group (Z15,+), or a residue class of 15 modulo some integer, or the integer 15? you might mean any of these things.

    my guess is you mean the principal ideals generated by 15 and 243, in which case you want the smallest ideal of (presumably Z) that contains (15) and (243). wouldn't any element of that have to be of the form 15a + 243b? can you think of a simpler description?
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  3. #3
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    Re: Elements generated by...

    Sorry. This is cycle notation. (15) is the transposition that sends 1 to 5 and 5 to 1. (243) is a cycle that sends 2 --> 4, 4 --> 3, and 3 --> 2.
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  4. #4
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    Re: Elements generated by...

    i see. so we are talking about S5, i hope. these are disjoint cycles, so they commute. so we have 6 elements at best:

    (1 5)^k(2 4 3)^m, k = 0,1 m = 0,1,2.

    calculate powers of (1 5)(2 4 3), how many do you get?
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  5. #5
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    Re: Elements generated by...

    That's kind of what i thought. I think...

    So, <(15), (243)> = { (1), (15), (243), (234), (15)(243), (15)(234) } Right?
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  6. #6
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    Re: Elements generated by...

    pretend i'm a skeptical professor, and convince me that any subgroup of S5 that contains (1 5) and (2 4 3) has to contain those 6 elements.
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