Page 1 : T. Y. B. Sc. (Group Theory) Practical's, Practical No. 1 - Binary Operations and Isomorphic Binary Structures, Determine whether the definition of * does give a binary operation on the, following set., 1., On Z+, define * by letting a * b= a - b., Determine whether the following given map o is an isomorphism of the, first binary structure with the second., 2., a) (Z, +) with (Z, +) where @(n) =n+1 for n E Z, b) (Q +) with (Q +) where ø(x) = x/2 for x E Q, The map o : Z→Z defined by ø(n) = n+ 1 for n EZ is one to one and, onto Z. Give the definition of a binary operation * on Z such that o is an, isomorphism mapping, (Z, +) onto (Z, *), 3., The map o : Q→Q defined by ø(x) = 3xr -1 for x E Q is one to one and, onto Q. Give the definition of a binary operation * on Q such that o is an, isomorphism mapping, (Q. +) onto (Q, *), 4., Practical No. 2: Groups and Subgroups, Determine whether the following binary operation * gives a group, structure on the given set. If no group results, give the first group, axiom in the order that does not hold., 1., Let * be defined on Z by letting a * b= ab., Show that if G is a finite group with identity e and with an even, 2., number of elements, then there is a + e in G such that a * a = e., Which of the following groups are cyclic? For each cyclic group, list, all the generators of the group., G1 = (Z, +), Find the order of the following cyclic subgroup of the given group, generated by the indicated element., The subgroup of Z4 generated by 3, Let H be a subgroup of a group G. For a, be G, let a - b if and only, if ab-1 E H. Show that - is an equivalence relation on G., 3., 4.

Page 2 : Practical No. 3: Cyclic Groups and Groups of Permutations, Either give an example of a group with the property described, or, explain why no example exist for the following statement, a) A finite group that is not cyclic, b) An infinite group that is not cyclic, c) A cyclic group having only one generator, If o -(134562), ,1=(1243)(56) and u =(15)(34) are permutations in, S6 then compute the following, a) τσ b) σ2τ, Show by an example that every proper subgroup of a nonabelian, group may be abelian., 1., 2., c) |< o >| d) µ', 100, e) µo?, 3., Find the maximum possible order for an element of S,for the given, value of n., 4., n = 5 b) n = 6, Practical no. 4: Cosets and Theorem of Lagrange, Find the all cosets of subgroup H of group G., 1., a) G=Z and H =4Z, b) G=Z12 and H=< 2 >, Let G be a group of order pq, where p and q are prime numbers., 2., Show that every proper subgroup of G is cyclic., Show that a group with at least two elements but with no proper, nontrivial subgroups must be finite and of prime order., 3., For the following give an example of the desired subgroup and, group if possible. If impossible, say why it is impossible., a) A subgroup of an abelian group G whose left cosets and right, cosets give different partitions of G., b) A subgroup of a group G whose left cosets give a partition of G, into just one cell., 4.

Page 3 : Practical 5: Direct Products and Homomorphisms, Find the order of the given element of the direct product., 1., a) (2, 3) in Z6 × Z15, b) (8, 10) in Z12 × Z18, Compute the indicated quantities for the given homomorphism p., 2., Ker(@) and ø(25) for @ : Z- Z7 such that @(1) = 4, Show that if G, G', and G" are groups and if o : G→G° and, y: G' → G" are homomorphisms, then the composite map, yo : G→ G" is a homomorphism., 3., The sign of an even permutation is +1 and the sign of an odd, permutation is -1. Observe that the map sgn, : S,→ {1,-1} \, defined by sgn,(6) = sign of o is a homomorphism of S, onto the, multiplicative group {1, -1} What is the kernel?, 4., Practical No. 6: Factor Groups and Simple Groups, Find the order of the following factor groups, a) (Z4 x Z12)(< 2> x <2>), b) (Z2 x Z4)<(1, 1)>, Give the order of the element in the factor group, 1., a) 5 + <4> in Z12/<4>, b) (3, 1) + < (1, 1) > in (Z4 × Z4)/< (1, 1) >, Show that if H and N are subgroups of a group G, and N is normal, in G, then HN N is normal in H. Show by an example that HN N, need not be normal in G., 2., 3., Show that if a finite group G contains a nontrivial subgroup of, index 2 in G, then G is not simple., 4.