8.6 An Application to Combinatorics
1. Let H act on G by h · x = hx for x 
G, h H. Then the orbits H · x = Hx are right cosets. Given
G, h H. Then the orbits H · x = Hx are right cosets. Given
Thus the Cauchy-Frobenius lemma gives the number of cosets as 
3. a. If the vertices are labeled as shown, the group G ⊆ S3 of motions is G = {ε, (23)}. Hence |F(ε)| = q3 and |F((23))| = q2, so the number of orbits is 
by Theorem 2.
by Theorem 2.
4. a. By Example 3 §2.7, the group of motions of the tetrahedron is A4. Now |A4| = 12 and A4 consists of ε, eight 3-cycles, and (1 2)(3 4), (1 3)(2 4) and (1 4)(2 3). Hence |F(σ)| = q2 for all σ A4 except σ = ε. Hence the number of colorings is 
by Theorem 2.
A4 except σ = ε. Hence the number of colorings is by Theorem 2.
5. a Label the top and bottom as 1, 2, and the sides 3, 4, 5, 6 as ...
Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
