Let be a positive prime number. A p-group is a group in which every element has order equal to a power of A finite group is a -group if and only if its order is a power of
There are many common situations in which -groups are important. In particular, the Sylow subgroups of any finite group are -groups. Since -groups have many special properties, they are easier to understand and classify than arbitrary groups, but they are useful since they are, in a sense, the "building blocks" for arbitrary groups via the Sylow theorems.
The first Sylow theorem can be used to show the following claim from the introduction:
A finite group is a -group if and only if its order is a power of
First, suppose that the order of a finite group is a power of By Lagrange's theorem, the order of any element divides the order of the group; but the only divisors of a power of are powers of so the group is a -group.
On the other hand, suppose that a group is a -group. Suppose its order is not a power of then there is another prime which divides its order. By the first Sylow theorem, there is a nontrivial Sylow -subgroup, whose order is a power of But then the order of any nontrivial element is a power of which is not a power of which is a contradiction. This proves the other direction of the equivalence.
An elementary application of the class equation gives the following nontrivial fact about finite -groups:
The center of a nontrivial finite -group is nontrivial.
Recall that the center of a group is the set of elements which commute with everything in for all
This follows from the second form of the class equation: for some elements not in the center. The point is that the terms in the sum cannot equal because if they did, would be all of and so would be in the center. If is a -group, then those terms have to be divisible by (powers of actually), and so taking both sides of the equation modulo gives So divides and hence it is not equal to 1.
Every -group is solvable.
First there is a basic fact:
If and are solvable, so is
To see this fact, note that a composition series corresponds to a series via the third isomorphism theorem; and the composition factors are the same. Concatenating with the series that exhibits the solvability of gives a series that exhibits the solvability of
Now the theorem follows by an easy induction. Suppose it is true for all -groups of order < (The base case, the trivial group, is trivially solvable.) If is abelian, then it is clearly solvable (since the composition factors are simple and abelian, hence cyclic of prime order). Otherwise, both and are smaller -groups than (since is nontrivial by the previous theorem), and hence must be solvable by the inductive hypothesis. Then is solvable by the fact proved in the previous paragraph.
The nontriviality of the center is the basis for many inductive arguments. The following problem gives another example.
Let be a group of order where is prime. Consider the following statements:
I. has a normal subgroup of order
II. has a normal subgroup of order
III. has a normal subgroup of order
IV. has a normal subgroup of order
How many of the statements I-IV is/are always true for any of order ?
Terminology: The order of a finite group is the number of elements in the group.
Every group of prime order is isomorphic to the cyclic group
Let be a nontrivial element of The order of is a nontrivial divisor of so it must be It is not hard to check that the function defined by is an isomorphism.
Every group of order where is a prime, is abelian. There are two such groups: and
Let be a group of order Every subgroup has order either or by Lagrange's theorem. Suppose is not abelian. Since is nontrivial and not all of it must have order Let be an element not in Then contains the center, but also contains So it is strictly larger than a subgroup of order The only possibility for its order is but then everything commutes with and is in the center; contradiction.
The classification of abelian groups of order follows from the general classification theorem for abelian groups.
Groups of order are harder to classify. Here is the statement of the result, without a complete proof.
For any prime there are five groups of order Three of these are abelian: When the two nonabelian groups are the dihedral group of rotations of the square, and the quaternion group When is odd, the two nonabelian groups are and (In both cases, the group operation is matrix multiplication.)
(1) When For odd they are not isomorphic: every nontrivial element of has order but has elements like of order
(2) and both have one element of order and elements of order but they are not isomorphic. So just knowing the orders of all the elements in a group is not enough to determine the group up to isomorphism.
(3) the dihedral group of a square, is the group of rotations and reflections of the square. is the group whose elements are with multiplication given by