midterm

MIDTERM

e stands for identity element.

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

group Z36. (2) Let G be a group of order n, a be an element of G (arbitrary element). Prove that there

exists a positive integer m with 0 < m  n such that am = e. (Hint: If not, write down the group H generated by a, i.e H =< a >. Then you can try to argue |H| > |G|, which is absurd.)

(3) Let set G be the Cartisian product of G1 ⇥G2, where (G1, ⇤1) and (G2, ⇤2) are 2 groups. Define a binary operation ⇤ on G by (a, b) ⇤ (c, d) = (a ⇤1 c, b ⇤2 d). Show that (G, ⇤) is group.

If (G, ⇤) is a commutative group, is G1 a commutative group? If yes, prove it. If not, give a counterexample.

(4) Let H := {A 2 Mn⇥n|A = A�1, det(A) 6= 0} be a set, ⇤ be matrix multiplication. Is the set (H, ⇤) a group? Explain it.

(5) Assume that G is a group such that for all x 2 G, x ⇤ x = e. Prove that G is an abelian group.

Date: May 2, 2021.

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