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Math Help - Number of Elements of Order 4

  1. #1
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    Number of Elements of Order 4

    True of false - for any group G, the number of elements x belongs to G such that x^4 = e is at most 4.

    I found on another forum the following solution but many steps were skipped...

    "Clearly e^4 = e.

    x^5 is an element such that (x^5)^4 = e.

    So are x^10 and x^15.

    But these are just 4 elements.

    So clearly there is no counterexample with a cyclic group.

    Now you try examples with non-cyclic Abelian groups, and Non-Abelian groups and see what you come up with..."

    what actually is C_20??and i cant think of an example of any non-cyclic abelian groups can anyone please talk me through where is method is going, preferably step by step??thanksss
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  2. #2
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    What about \mathbb{Z}_2^5, i.e., 5-tupes of 0, 1 under addition?

    People in the Algebra forum may be better equipped to answer this question. I would ask a moderator to move this thread.
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  3. #3
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    here is a non-abelian group of order 8, the 4th dihedral group D4 = {1,r,r^2,r^3,s,rs,r^2s,r^3s} with r^4 = s^2 = 1, sr = r^3s.

    we have the subgroup of order 4, {1,r,r^2,r^3} and clearly, for every element x in that subgroup, x^4 = 1.

    (1^4 = 1, r^4 = 1, (r^2)^4 = (r^4)^2 = 1^2 = 1, (r^3)^4 = (r^4)^3 = 1^3 = 1).

    so there are 4 elements right there. but s^4 = (s^2)^2 = 1^2 = 1, so s is a 5th element whose 4th power is 1.

    when looking for counter-examples in groups, non-abelian groups of small order often work well.
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  4. #4
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    thank you once again for such detailed explanations...i do understand stuff much better like this compared to when they skip a lot of the steps~~thank you!!!!

    Quote Originally Posted by Deveno View Post
    here is a non-abelian group of order 8, the 4th dihedral group D4 = {1,r,r^2,r^3,s,rs,r^2s,r^3s} with r^4 = s^2 = 1, sr = r^3s.

    we have the subgroup of order 4, {1,r,r^2,r^3} and clearly, for every element x in that subgroup, x^4 = 1.

    (1^4 = 1, r^4 = 1, (r^2)^4 = (r^4)^2 = 1^2 = 1, (r^3)^4 = (r^4)^3 = 1^3 = 1).

    so there are 4 elements right there. but s^4 = (s^2)^2 = 1^2 = 1, so s is a 5th element whose 4th power is 1.

    when looking for counter-examples in groups, non-abelian groups of small order often work well.
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by bryan06 View Post
    what actually is C_20??
    C_20 is the cyclic group of order 20. It is often denoted by Z/20Z, or Z_20.
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