Number of Groups of A Given Order
Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are. Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. If n is the square of a prime, then there are exactly two possible isomorphism types of group of order n, both of which are abelian. If n is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order n, and the number grows very rapidly as the power increases.
Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when q < p are primes with p-1 not divisible by q. For a necessary and sufficient condition, see cyclic number.
If n is squarefree, then any group of order n is solvable. A theorem of William Burnside, proved using group characters, states that every group of order n is solvable when n is divisible by fewer than three distinct primes. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order n is solvable when n is odd.
For every positive integer n, most groups of order n are solvable. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the classification of finite simple groups. For any positive integer n there are at most two simple groups of order n, and there are infinitely many positive integers n for which there are two non-isomorphic simple groups of order n.
Read more about this topic: Finite Group
Famous quotes containing the words number of, number, groups and/or order:
“He is the richest man who knows how to draw a benefit from the labors of the greatest number of men, of men in distant countries, and in past times.”
—Ralph Waldo Emerson (1803–1882)
“Strange goings on! Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight. What he did was butter a piece of toast. We are too familiar with the language of action to notice at first an anomaly: the ‘it’ of ‘Jones did it slowly, deliberately,...’ seems to refer to some entity, presumably an action, that is then characterized in a number of ways.”
—Donald Davidson (b. 1917)
“Writers and politicians are natural rivals. Both groups try to make the world in their own images; they fight for the same territory.”
—Salman Rushdie (b. 1947)
“Yet whatever gains I had, these I have come to regard as loss because of Christ. More than that, I regard everything as loss because of the surpassing value of knowing Christ Jesus my Lord. For his sake I have suffered the loss of all things, and I regard them as rubbish, in order that I may gain Christ and be found in him, not having a righteousness of my own that comes from the law, but one that comes through faith in Christ, the righteousness from God based on faith.”
—Bible: New Testament, Philippians 3:7-9.