Theorem (Frobenius) - Berliner Sitzungsberichte, 1895, p. 984
If n divides the order of a group, then the number of elements in the group whose orders divide n is a multiple of n.
Proof: Let G be a group of smallest order g for which the Theorem fails. By this assumption the Theorem is true for all groups H with We call G a minimal counterexample. We proceed to contradict the minimality of G, and thus conclude that such G, in fact, does not exist.
For the group G, let n be the largest factor of g for which the
Theorem fails. Since the Theorem is clearly true for
n=1 and n=g, it follows that
1<n<g. Let Nm denote the number of elements
in G whose order divides m. Furthermore, let p be a prime which
divides g. We have
where Nn' denotes the number of elements in G whose order divides
np but not n. We aim to show that
Since by the
we conclude the
proof if we show that
then n divides the difference
Let order of x divides np but does not divide ; Write with (p,s)=1, that is, p and s are relatively prime. Observe that if then divides the order of x. To show that divides we first show that divides then show that s divides (Note also that if we are done.)
1. Let Then all the generators of <y1> are in A; denote this set by A1. Select Again there are generators of <y2>, all of them in A; call this set A2. Note that , for if then <y1>=<y>=<y2>, a contradiction. Since A is finite we have (disjoint union), and since divides Ai for all i, we conclude that divides and hence divides
2. (a) If
then x can be written uniquely as x=yz,
with y and z commuting and
Indeed, the cyclic
group <x> can be written as a direct sum <x>=YZ, where Y is
a subgroup of <x> of order
and Z is a subgroup of <x> of
order t. Thus x=yz, with
as stated. Furthermore,
since y and z are in <x>, they are both powers of x.
As to uniqueness, if x=yz, with y and z as above, then Since we conclude that <y,z>=<x>. Thus y and z are in <x> and uniqueness follows from the internal direct sum decomposition of <x>=<y><z>. We call y the p-constituent of x.
(b) For with let is the p-constituent of Note that if y1 and y2 are two distinct elements of order in A, then indeed, this follows from the uniqueness of the p-constituent established in (a) above. We can now conclude that the subsets Cy partition A.
(c) Let Denote by Ky the centralizer of y in G. Clearly Let and let d=gcd(r,s), that is, d is the greatest common divisor of r and s. Since r<g, by the inductive assumption there exist a multiple of d, say cd, elements in Ky/<y> whose orders divide d. Let be such a coset of <y> in Ky. We can represent in the form z<y>, with a divisor of d. [Indeed, , and therefore z1t=yi, for some i. We seek j such that z=z1yj satisfies 1=zt=z1tyjt=yiyjt=yi+jt. Since such a j exists; take j=-it-1 (mod ] Then Each element of Ky/<y> whose order divides d produces in this manner an element of Cy. Thus
(d) Consider the disjoint union taken over all the conjugates of y in G. We assert that s divides We have . Since g is divisible by s and r, it follows that gd is divisible by rs. Hence s is a divisor of Since s and are relatively prime, s must also be a factor of . In view of (b) above, this shows that s divides This ends the proof.
By making use of the Möbius function we can obtain an expression for the number of elements whose order is exactly n in terms of the number of elements whose orders divide the divisors d of n. Specifically, denote by f(d) the number of elements whose orders are exactly n. Let h(d) denote the number of elements whose orders divide d. Clearly we have, Möbius inversion now yields where is defined to be 0, unless n/d is a product of k distinct primes, in which case it equals (-1)k. Frobenius' Theorem informs us that h(d)=m is a multiple of d. We thus conclude as follows:
* If n divides the order of a group, then the number of elements of order exactly n is equal to . Here the sum is over only those divisors d of n with the property that n/d is a product of any number (k, say) of distinct primes. Furthermore, m is the number of elements whose orders divide d.