{'fraction_non_alphanumeric': 0.0681265206812652, 'fraction_numerical': 0.01824817518248175, 'mean_word_length': 4.0182926829268295, '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}}
Generalizations of certain fundamental results on finite groups

In this short note, we synthesize various fundamental results of finite groups and their proofs to obtain a generalization of J. G. Thompson's fundamental A x B Lemma (cf. [5, Theorem 5.3.4]), a generalization of a fundamental result of H. Bender (cf. [2, Satz] and [4, Theorem 3.1]) and a generalization of an important result of N. Blackburn (cf. [3, Theorem]). Our notation is standard and tends to follow the notation of [5]. In particular, p denotes a prime integer and all groups that we consider are finite. Our first four results concern the following basic situation. V is a p-group acted on by the group G where G = KP with K = O,(G) and P E Sylp(G). We begin by observing that the proof of [2, Satz] yields a generalization of [5, Theorem 5.2.4].