State, with reasons, whether the following statements are True or False (a) Any

Question

State, with reasons, whether the following statements are True or False (a) Any subgroup H subset G of a group G is the kernel of a suitable group homomorphism phi: G rightarrow G_1. (b) If m > 1, n > 1 are natural numbers with m > n, there is a group G of order m and a set S with n elements such that G operates (c) For every n elementof N, n > 2, there are infinitely many irreducible polynomials of degree n over Q (d) Let R_1, R_2: G rightarrow GL_n (R) be two irreducible representations of the finite group G. Then, their characters chi_R_1 and chi_R_2 are orthogonal only if R_1 and R_2 are inequivalent. (c) The fields Q(pi) and Q(pi^2) are isomorphic.

Explanation / Answer

a) True, consider coset and the canonical map.

b) The statement is incomplete.

c) True, consider integral multiples of roots of unity as roots of polynomials.

d) False as pi is not an element of 2nd extension(as pi is not algebraic).

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