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Let a be an element of.order n in a group and k be a positive integer .prove that |ak| = n/(gcd (n,k))
Consider A, B, and C three points of the plan. Show that det(AB, AC) = det(BC, BA) = det (CA, CB)
a) Using a geometrical approach
b) Using the property of antisymmetry of the determinant
Find an Dedekind finite ring that is not IC (other than examples 5.10 & 5.12 given in K.R. Goodearl - Von Neumann Regular Rings)
Prove or disprove: If G is a finite group and some element of G has order equal
to the size of G, then G is cyclic.
(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]
(a) show that (G,∗) is a group.
(b) Is G abelian? if not, find its center.
(c) Give an automorphism of G.
