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Prove that in any abelian group G, the set {g/g²=e} is a subgroup of G. Does this result remain true if G is not abelian? Justify your answer.
Let X be a path-connected space and suppose that every map f: S^1 --> X is homotopically trivial but not necessarily by a homotopy leaving the base.
Let G be any non-Abelian group of order 6. By Cauchy's theorem, G has an element, a, of order 2. Let H = a, and let S be the set of left cosets of H.
Show that the conclusion does not follow if we assume the relation (a.b)i = aibi for just two consecutive integers.
Find order of all elements in s3 , where s3 is the symmetric set of permutations of degree 3.
If G is a group of even order, prove it has an element a ? e satisfying a2 = e.
Let G be a nonempty set closed under an associative product, which in addition satisfies Then G is a group under this product.
Consider the groups Z3 x Z3 and Z9. These are each "integer groups" of order 9. Are they isomorphic or not? Give an explicit reason.
Prove that a group of order p^2, where p is a prime number, must have a normal subgroup of order p.
Suppose H is the only subgroup of order O(H) in the finite group G. Prove that H is a normal subgroup of G.
Let G = UT (n,F) be the set of the upper triangular n x n matrices with entries in a field F with p elements and 1's on the diagonal.
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.