Prove that in a group, (a-1)¯' = a for all a.

Let G be a group and a e G such that o(a) =n < oo. Show that a = a' if and only if k =l mod n.

Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D %3D where a = (1 2 3 4) and b = (2 4). Compute (ab)(ab) Compute (ab)(a²b) Compute (ab)(a³b)

is H(a+bi)=1+i a homomorphism and why?

Create steps along with justifications to verify that in a group system (G, +) the following property holds:For any two elements from G, ‘a’ and ‘b’, -(a + b) = (-b) + (-a). In other words, the claim is that (-b) + (-a) plays the role of the inverse of a + b. Create steps to show that the element (-b) + (-a) does, in fact, play the role of an inverse to the element a + b, i.e., show that:i. (a + b) + ( (-b) + (-a) ) = e, where e represents the identity in the group; andii. ( (-b) + (-a) ) + (a + b) = e.

Show that the frieze group F6 is isomorphic to Z ⊕ Z2.

Define X = R \ {k} and define ⋆ to be the operation such that x + y = = (x − k) (y − k) + k. Check each of the four axioms of a group (closure, associative, identity, inverse). Which of them hold? Is (X, *) a group?

7. If a is a group element and a has infinite order, prove that a™ ‡ an when m # n.