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What are the situations in which any, or particular type group of order n, is abelian ?

For example:

Group of order $p^2$ is abelian. where p is prime.

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any group of order $p$ is cyclic (where p is prime), and therefore abelian (any cyclic group is abelian, but not all abelian groups are cyclic). Yes, as @reader suggests, I think you need to clarify what you're asking. Are you asking for a characterization of all abelian groups? –  amWhy Nov 2 '12 at 16:29
    
It's unclear what you mean by "any, or particular type group" means. Do you mean to ask, "For what values of $n$ do we know that all groups of order $n$ are abelian?" –  Thomas Andrews Nov 2 '12 at 16:29
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I think the necessary and sufficient conditions on $n$ are that for all primes $p,q$ dividing $n$: (i) $p^3$ does not divide $n$; (ii) $q$ does not divide $p-1$; and (iii) if $p^2 \mid n$ then $q$ does not divide $p^2-1$. Hope that's right! –  Derek Holt Nov 2 '12 at 16:30
    
Thomas Andrews Sir !!! you are right, I want that. –  ram Nov 2 '12 at 16:39
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1 Answer 1

up vote 8 down vote accepted

Every group of order $n$ is abelian if and only if $n$ has prime factorization of the form $$n = p_1^{a_1} p_2^{a_2} \ldots p_t^{a_t}$$

where $a_i \leq 2$ for all $i$ and $\gcd(p_i^{a_i} - 1, p_j) = 1$ for all $i$ and $j$.

This an old result (1905) and due to L. E. Dickson. Reference:

Dickson, L. E. Definitions of a group and a field by independent postulates, Trans. Amer. Math. Soc. Vol. 6, No. 2, 198-204 (1905).

You might be interested in a similar problem for other group properties besides being abelian. Similar characterizations are known for those $n$ for which every group of order $n$ is

  • cyclic
  • abelian
  • nilpotent
  • nilpotent of class at most $c \in \mathbb{Z}_+ \cup \{\infty\}$
  • solvable
  • supersolvable
  • metabelian
  • metacyclic
  • etc..

Worth mentioning is the case of cyclic groups, which is a cute result: every group of order $n$ is cyclic if and only if $\gcd(n, \varphi(n)) = 1$, where $\varphi$ is the Euler totient function. Also, some of these are far from trivial: characterizing those $n$ for which every group is solvable requires Thompson's classification of minimal simple groups. The result is also needed in the case of metabelian groups.

A good starting point for studying the easier results (cyclic, abelian, nilpotent) is the following paper by Pakianathan and Shankar:

Pakianathan, Jonathan. Shankar, Krishnan. Nilpotent Numbers. Amer. Math. Monthly, Vol. 107, No. 7, 631-634 (2000).

References for the cyclic case can be found in this question. See also the answer by Pete L. Clark in here.

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Can't you rewrite that as $\gcd(p_i^{a_1}-1,n)=1$ for all $i$? –  Thomas Andrews Nov 2 '12 at 18:22
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