Paper 3, Section II, D

Groups | Part IA, 2021

(a) Let xx be an element of a finite group GG. Define the order of xx and the order of GG. State and prove Lagrange's theorem. Deduce that the order of xx divides the order of GG.

(b) If GG is a group of order nn, and dd is a divisor of nn where d<nd<n, is it always true that GG must contain an element of order dd ? Justify your answer.

(c) Denote the cyclic group of order mm by CmC_{m}.

(i) Prove that if mm and nn are coprime then the direct product Cm×CnC_{m} \times C_{n} is cyclic.

(ii) Show that if a finite group GG has all non-identity elements of order 2 , then GG is isomorphic to C2××C2C_{2} \times \cdots \times C_{2}. [The direct product theorem may be used without proof.]

(d) Let GG be a finite group and HH a subgroup of GG.

(i) Let xx be an element of order dd in GG. If rr is the least positive integer such that xrHx^{r} \in H, show that rr divides dd.

(ii) Suppose further that HH has index nn. If xGx \in G, show that xkHx^{k} \in H for some kk such that 0<kn0<k \leqslant n. Is it always the case that the least positive such kk is a factor of nn ? Justify your answer.

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