Mini Quiz (13. November)

1. Suppose G is a group, |G|=24 and G=< a >. Find the order of .

Answer: 12

Justification. Since G=< a >, the elements of G are all powers of a. Since |G|=24, it follows that a²⁴ is the identity and no smaller n will make identity. So is the identity and 12 is the order of .

2. Suppose G=< a > and H=< b >. Find a finite set generating G x H.

Answer: (a,e_H), (e_G,b) generates G x H.

Justification. Straightforward.

3. is a finite group and H is a subgroup of G.

True or False:

(a) .

TRUE (G is the union of all right cosets).

(b) If then .

FALSE (Many counter examples.)

(c) such that |H|.m=n.

TRUE (Lagrange)

(d) If , then

TRUE (Otherwise, after multiplying by the inverse of h on the left, we would get that g is in H)

(e) is a divisor of n.

TRUE (Lagrange).