Question

Q15

(14 / Suppose that Gl ะ G2 and Hi ex H2. Prove that GI E HIN G-DH2. State the general case. If G H is cyclic, prove that G and H are cyclic. State the (1 eneral case. h Zso D Zso. find two subgroups of order 12. If r is a divisor of m and s is a divisor of n, find a subgroup of Zm Z, isomorphic to Z, ®Z,

Answer 1

First we will show that every subgroup of a cyclic group is cyclic. Then from that theorem we derive the above result.

First we will show that every subgroup of a cyclic group is cyclic. Proof: let G be cyclic, G = (a) and K be subgroup of G then a^m elementof K for m. let t be least purifier integer such that a^t elementof K. claim: K = (a^t) clearly, (a^t) subsetequalto K now, let b elemebntof K. then b elementof G. so, b = a^n for n. by division algorithm, m = t q + r for 0 lessthanorequalto r < t now, a^t elementof K Rightarrow a^-t elementof K as K is a subgroup so, (a^n) (a^-t)^q elementof K Rightarrow a^n - t q elementof K Rightarrow a^r elementof K. \r\n but 0 lessthanorequalto r < t and t is least purifier with a^t elementof K implies r = 0. Therefore n = t q and a^n = (a^q)^t = (a^t)^q elementof (a^t) therefore K subsetequalto (a^t) therefore K = (a^t) and