4. Let (G, *) be a group of order 231 = 3 × 7 × 11 and H€ Syl₁₁(G), KE Syl, (G). Prove that (a). HG and KG. (b). G has a cyclic subgroup of order 77.
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- Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40
- Prove that each of the following subsets H of GL(2,C) is subgroup of the group GL(2,C), the general linear group of order 2 over C a. H={ [ 1001 ],[ 1001 ],[ 1001 ],[ 1001 ] } b. H={ [ 1001 ],[ i00i ],[ i00i ],[ 1001 ] }34. Suppose that and are subgroups of the group . Prove that is a subgroup of .Find two groups of order 6 that are not isomorphic.
- 11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.