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Math 417 – Homework XI: July 24, 2007
2.38(i). Compute the order, inverse, and parityof
α=(1 2)(4 3)(13542)(1 5)(1 3)(2 3).
Using the algorithm, the complete factorization in S5of αis (154)(2 3), so
its order is lcm{2,3} = 6; its inverse is α−1=(1 5 4)−1(2 3)−1=(1 4 5)(2 3)
(since disjoint cycles commute, we do not have to change the ordering of the
factors); αis odd, being the product of an even (the 3-cycle) and an odd (the
tranposition).
2.38(ii). What are the respective orders of the permutations in Exercises 2.22
and 2.28?
The complete factorization of αin Exercises 2.22 is (1 9)(2 8)(3 7)(4 6)(5),
and so αhas order 2, for it is a product of disjoint transpositions.
Exercise 2.28 defines f: {0,1,2,...,10}→{0,1,2,...,10}by
f(n)=the remainder after dividing 4n2−3n7by 11.
is a permutation. Here is the two-rowed notation for f:
0 1 2 3 4 5 6 7 8 9 10
0 1 6 9 5 3 10 2 8 4 7 .
The complete factorization is f=(0)(1)(2 6 10 7)(3 9 4 5)(8), and so fhas
order 4. Since a 4-cycle is odd, we see that fis odd ×odd =even. Finally, the
inverse of fis (2 6 10 7)−1(3 9 4 5)−1=(7 10 6 2)(5 4 9 3).
2.55.Give an example of two subgroups Hand Kof a group Gwhose union
H∪Kis not a subgroup of G.
If G=S3, then H= {(1 2), (1)}, and K= {(1 3),(1)}are subgroups. But
H∪Kis not a subgroup of G, for (1 2)(1 3)=(1 3 2) /∈H∪K.
2.46.If Gis a group with an even number of elements, prove that the number
of elements in Gof order 2 is odd. In particular, Gmust contain an element of
order 2.
Gis the disjoint union of Xand Y, where
X= {g∈G:g−16= g}and Y= {g∈G:g−1=g}.
