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if H and G are two isomorphic groups then prove that Atu(H) is isomorphic to Atu(G).

The set of discontinuous functions from [1,0] to R form a ring with respect to pointwise addition and multiplication. Is this statement true or false, justify your answer.

The characteristic of a finite field is zero. Is this statement true or false, justify your answer.

If a ring has a unit, then it has only one unit. Is this statement true or false ,justify your answer.

If σ ∈ Sn (n ≥ 3 ) is a product of an even number of disjoint cycles, then sign (σ) = 1. Is this statement true or false, justify your answer.

If R is a ring and I is an ideal of R , then xr = rx ∀ x ∈ I and r ∈ R . Is this statement true or false, justify your answer .

Every element of Sn has order at most n . Is this statement true or false, justify your answer.

If G is a group and H∆G , then | G : H |= 2 . Is this statement true or false, justify your answer.

If G is an abelian group, then G is cyclic. Is this statement true or false . Also justify your answer.

If G1 and G2 are groups, and f : G1→G2 is a group homomorphism, then

o(G1) = o( G2). Is the statement true or false ? Also justify.

o(G1) = o( G2). Is the statement true or false ? Also justify.