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Math Help - How to check group of given order cyclic or not?

  1. #1
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    How to check group of given order cyclic or not?

    if we've given any order of group then how can we check that is it cyclic or not example how can we check group of order 255 and 465 and 3000 order cyclic or not.Please help
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    Quote Originally Posted by reflection_009 View Post
    if we've given any order of group then how can we check that is it cyclic or not example how can we check group of order 255 and 465 and 3000 order cyclic or not.Please help
    If |G| = pq with q>p primes and q\not \equiv 1 (\bmod p) then G\simeq \mathbb{Z}_{pq}. Otherwise q\equiv 1(\bmod p) then it is not true.

    Now 255 = 5\cdot 51 there the group is not necessarily cyclic.

    Note 465 = 3\cdot 5\cdot 31 let H be a non-cyclic group of order 3\cdot 31. Then | \mathbb{Z}_5 \times H| = 465 and \mathbb{Z}_5 \times H is not cyclic.

    If |G| = 2n where n\geq 3 then we can construct a non-abelian group D_n - the dihedral.
    Therefore a group of order 3000 is not necessarily cyclic.
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    But 255 is is cyclic i've read in book.
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