{'fraction_non_alphanumeric': 0.035386631716906945, 'fraction_numerical': 0.009174311926605505, 'mean_word_length': 4.288683602771362, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, '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}}
TWO RESULTS ABOUT FINITE GROUPS

I would like to discuss two results about finite groups: (0) All finite groups of odd order are solvable. (F) A finite group is nilpotent if it admits a fixed point free automorphism of prime order. Walter Feit and I proved (0) after a prolonged joint effort [5]. A critical special case of (0) was proved by Walter Feit, Marshall Hall, Jr. and me [4]. A slightly stronger result than (F) is in the literature [16]. It was Suzuki who took the first big step toward the proof of (0), for he showed that the group G is solvable if it is of odd order and if in addition the centralizer of each non-identity element of G is abelian [15]. The proof of this apparently very special case removed one of the major stumbling-blocks in the proof of (0). Suzuki's proof contains an application of exceptional characters. Exceptional characters were discovered by Brauer and Suzuki and their simplicity and power have been of great help in recent work in finite groups. Suzuki's proof also presents a structure for the proof of (0). This structure is very easy to describe; it has two parts: (1) Determination of the maximal subgroups. (2) Exploitation of Frobenius reciprocity. The special case of (0) alluded to consists simply in replacing "abelian" by "nilpotent" in Suzuki's result. A skirmish with ^-groups and a modification of one character argument mark the only divergence from Suzuki's proof. The proof of (0) involves a large number of technical problems and a great deal of case analysis. Proving (0) by induction on the order of G, it can be assumed that G is simple and that every proper subgroup of G is solvable. In the determination of the maximal subgroups of G, a much more serious contact with solvable groups is required than is customary. An initial analysis, to which I will return, shows that the maximal subgroups of G are of two types. Type 1 consists of those subgroups M which are split extensions of a nilpotent normal subgroup H by a group E with the property that E contains a subgroup E 0 of the same exponent as E such that non-identity elements of E 0 induce fixed point free automorphisms of H. Furthermore, the set of elements of E which have non-identity fixed points on H lies in a normal abelian subgroup of E. Type 2 consists …