{'fraction_non_alphanumeric': 0.04375, 'fraction_numerical': 0.011805555555555555, 'mean_word_length': 3.9689655172413794, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On a theorem of Frobenius: solutions of X n = 1 in finite groups

Given a finite group G and a positive integer n, we write fj(G) to denote the number of solutions in G to the equation x = 1. A celebrated theorem of Frobenius asserts that if n divides IGI, then n divides fj(G). There are numerous proofs of Frobenius' theorem in the literature (see, for example, [1] or Corollary 41.11 of [2] or Theorem 9.1.1 of [3]); some of these are representation theoretic and others are "elementary". The authors believe, however, that the proof presented below is the easiest yet to appear. This paper was written while the authors were guests of the Mathematics Research Section of the School of Mathematical Sciences at the Australian National University in Canberra. We thank the A.N.U. for its support and hospitality. In the following, the symbol "p" will always represent a prime integer, and "G" a finite group. If n > 0 is an integer, we shall write np to denote the largest power of p which divides n. For example 242 = 8. If g E G is any element, it is well known that there is a unique decomposition g = xy with x and y commuting, such that x has p-power order and y has order not divisible by p. This establishes a bijection between the elements of G and the ordered pairs (x, y) for which y has order prime to p and x E CG(y) has p-power order. We use this correspondence to prove the following lemma, which is the key to our argument.