Math 4124Monday, March 21March 21, Ungraded HomeworkProve that a group of order 175 is abelian.LetGbe the group of order 175=7·52. The number of Sylow 7-subgroups is congruentto 1 mod 7 and divides 25. The only possibility is 1, soGhas a normal Sylow 7-subgroup,call itA. Also the number of Sylow 5-subgroups is congruent to 1 mod 5 and divides 7.ThereforeGhas a normal Sylow 5-subgroup, call itB. SinceAandBare of prime or primesquared order, they are both abelian, in particularA≤CG(A). Furthermore by Lagrange’stheoremA∩B=1. SinceA,BG, we deduce that every element ofAcommutes with everyelement ofB(recall that one proves this by consideringaba-1b-1= (aba-1)b-1∈Bandsimilarly∈A, soaba-1b-1∈A∩B=1). ThusB≤CG(A). By Lagrange’s theorem|A|,|B|divide|CG(A)|, so|CG(A)|=175, consequently CG(A) =G. This means thatA≤Z(G).SimilarlyB≤Z(G)and we deduce that|A|,|B|divide|Z(G)|. It follows that Z(G)has ordera multiple of 7·25, which is only possible if Z(G) =G. This means that
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