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Prove that any element s in Sn which commutes with (1, 2 ...r) is of the form s = (1, 2,....., r)it where i = 0, 1, 2,.......,r, , t is a permutation.
Prove that a group of order 108 must have a normal subgroup of order 9 or 27.
Verify if the mappings defined is a homomorphism and in that case in which it is homomorphism.
Let G be a group and Z(G) , the centre of G, then G / Z(G) I(G), where I(G) is the set of all inner automorphisms of G.
What is the order of the product of the disjoint cycles of lengths m1 , m2, ......mk ?
Determine which are even permutations .Provide complete and step by step solution for the question.
If in a finite group G an element a has exactly two conjugates, prove that G has a normal subgroup N ? eG.
Find all 3-Sylow subgroups of or, Sylow 3-subgroups and 2-Sylow subgroup or, Sylow 2-subgroups of the symmetric group of degree 4, S4.
If x is an element of a group and x is of order n then the elements 1, x, x^2,...x^n-1 are distinct.
How to prove that the order of an element in Sn equals the least common multiple of the lengths of the cycles in its cycle decomposition.
Let G be a finite group with K is a normal subgroup of G. If (l K l, [G:K]) =1, prove that K is unique subgroup of G having order l K l.
Prove that if H is nontrivial normal subgroup of the solvable group G then there is a nontrivial subgroup A of H with A normal subgroup of G and A abelian.
If M is a finite abelian group then M is naturally a Z-module. Can this action be extended to make M into a Q-module?
Let G be finite and P be a Sylow p subgroup of G. Suppose the normalizer of P in G is a subset of H is a subset of G.
Prove that the following are equivalent ~ is an equivalence relation of a group G
If H and K are normal subgroups of a group G with HK = G. Prove that G/(H n K) = (G/H) x (G/K).
Let G be a group, let a, b be elements in G and let m and n be (not necessarily positive) integers. (a^n)^m= (a^mn)
Give an example of groups Hi, Kj such that H1xH2 is isomorphic to K1xK2 and no Hi is isomorphic to any Kj.
Prove that if G is a finite group, H subset of G that is closed with respect to the operation of G, Then every element of H has its inverse in H.
Prove that there cannot be a nilpotent group N generated by two elements with the property that every nilpotent group generated by two elements.
Let G be a nonempty finite set equipped with an associative operation such that for all a,b,c,d in G. Prove that G is a group.
Prove that the cyclic subgroup generated by a^m is the same as the cyclic subgroup generated by a^d, where d = (m,n).
Show that G is a group of order 8 and that G is isomorphic to the quaterunion group Q = {1, i, -1, -i, j, k, -j, -k }.
Prove that the subgroup of A4 generated by any element of order 2 and and any element of order 3 is all of A4.
If G is nonabelian and the order of G is 24 and G is isomorphic to H x Z_3, what are the possibilities for H up to isomorphism.