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PRACTICAL QUESTIONS

(1) Prove that every subgroup H of cyclic group G is a cyclic group. Find all generators of group Z32.

(2) Show that group Un (n th unit root) and group Zn are isomorphic. (3) Is all groups of order 6 commutative ? If not commutative, give an example. How about

groups of order 5? (Are they always commutative). (4) Let set G be the Cartisian product of G1 ×G2, where G1 and G2 are 2 groups. Define a

binary operation ∗ on G by (a, b) ∗ (c, d) = (ac, bd). Show that (G,∗) is group. If both G1 and G2 are commutative groups, show G is also a commutative group.

(5) Let H := {A ∈ Mn×n|A = AT , det(A) 6= 0} be a set, ∗ be matrix multiplication. Is the set (H,∗) a group? What if ∗ is the matrix addition?

(6) Assume that G is a group such that for all x ∈ G, x∗x = e. Prove that G is an abelian group.

Date: April 23, 2021.

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