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Let G be a nonabelian group of order 10 having golden ratio 1+√ 5 2 as the neutral element. Then: (a) Verify class equation for G. (b) Find all elements of Inn(G). (c) Verify that G/Z(G) ∼= Inn(G). (d) For some elements x, y, z ∈ G: verify the commutator identity: [xy, z] = [x, z][x, z, y][y, z]
(a) show that (G,∗) is a group.
(b) Is G abelian? if not, find its center.
(c) Give an automorphism of G.
nontrivial example. { same example by all students may lead to -ve marking }
(i) S4 is isomorphic to D12.
(ii) H ∼
= K if and only if Aut(H) ∼
= Aut(K).
(iii) Every action of the group G gives the same orbit space.
(iv) The isomorphism class of the multiplicative group of real numbers is non-empty.
(v) The converse of Cauchy’s theorem is true.
ln β
. Then:
(a) show that (G, ∗) is a group.
(b) Is G abelian? if not, find its center.
(c) Give an automorphism of G
Give an example of infinite ring of characteristic 2
(a) show that (G,∗) is a group.
(b) Is G abelian? if not, find its center.
(c) Give an automorphism of G.
Let G be a nonabelian group of order 10 having golden ratio 1+√5/2 as the neutral element. Then: (a) Verify class equation for G. (b) Find all elements of Inn(G). (c) Verify that G/Z(G) ∼ = Inn(G). (d) For some elements x,y,z ∈ G: verify the commutator identity: [xy,z] = [x,z][x,z,y][y,z]
If the following are true, give detailed proof. Otherwise, support your answer by a nontrivial example.
(i) S4 is isomorphic to D12.
(ii) H ∼ = K if and only if Aut(H) ∼ = Aut(K).
(iii) Every action of the group G gives the same orbit space.
(iv) The isomorphism class of the multiplicative group of real numbers is non-empty.
(v) The converse of Cauchy’s theorem is true.
Let G be the set of positive real numbers except 1. Define α∗β = αlnβ. Then:
(a) show that (G,∗) is a group.
(b) Is G abelian? if not, find its center.
(c) Give an automorphism of G.
Let G be the set of positive real numbers except 1. Define α∗β = αnβ. Then:
(a) show that (G,∗) is a group.
(b) Is G abelian? if not, find its center.
(c) Give an automorphism of G.
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