Chapter 8: Q15E-b (page 246)
Question:In Exercise 13-15, is a subgroup of G . Determine whether the given cosets are disjoint or identical.
15 ;
(b) and
The given cosets are identical.
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Let be a group that contains at least one subgroup of order . Let , where the intersection is taken over all subgroups of order . Prove that is a normal subgroup of .[Hint: For each , verify that , where the intersection is over all subgroups of order ; use Exercise 20 of Section 7.4.]
Prove that every homomorphic image of a metabelian group is metabelian.
Show by example that if M is a normal subgroup of N and if N is a normal subgroup of a group G , then M need not be a normal subgroup of G; in other words, normality isn’t transitive. [Hint: Consider and in
Question:In Exercise 13-15, K is a subgroup of G . Determine whether the given cosets are disjoint or identical.
14. ; K is the subgroup
(b) and .
Cayley’s Theorem 7.21 represents a group G as a subgroup of the permutation group A(G). A more efficient way of representing G as a permutation group arises from the following generalized Cayley’s Theorem. Let K be a subgroup of G and let T be the set of all distinct right cosets of K.
If , show that the map given by is a permutation set of the set T.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
