4. Show that the following are groups. (a) (Sn, , ), where Sn is the set of...

4. Show that the following are groups.

(a) (Sn, ⊕, Ô), where Sn is the set of binary strings of length n, ⊕ is the exclusive-or operation applied bitwise on such words, and Ô is the string of 0s of length n.

Can you use the fact of Exercise 2 for a quick argument that the operation ⊕ must be associative?

(b) (Zn + n, [0]n).

(c) (Z, +, 0). For which value of n does this "follow" from the previous item?

(d) (G, o, e), where

• G is the set of functions f : S → S, over some set S, that have a mathematical inverse f-1 : S → S.

• f o g is the function that maps all s ∈ S to f(g(s)), and

• e is the function that leaves all s ∈ S fixed.

(e) Which of the groups in (a)-

(d) are finite? Which ones are commutative?