The Symmetric, Alternating, and other Special Groups

Abstract

The importance of the symmetric group is obvious from the fact that any finite group, of order n, say, is a subgroup of S _n . The procedure for obtaining generators and relations for S _n can be carried over almost unchanged for a certain infinite group, first studied by Artin (1926); accordingly, we begin by describing this so-called braid group. The symmetric group (§ 6.2, p. 64) has a subgroup A_n, of index 2 (§ 6.3, p. 66), which is particularly interesting because, when n > 4, it is simple. In § 6.4, p. 67, we exhibit the groups S₃, A₄, S₄ and A₅ as members of the family of polyhedral groups (l, m, n),defined by

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaCa % aaleqabaGaamiBaaaakiabg2da9iaadofadaahaaWcbeqaaiaad2ga % aaGccqGH9aqpcaWGubWaaWbaaSqabeaacaWGUbaaaOGaamOuaiaado % facaWGubGaeyypa0Jaamyraaaa!425B!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${R^l} = {S^m} = {T^n}RST = E$$

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