College Algebra Exam Review 115

College Algebra Exam Review 115 - 2.5. COSETS AND LAGRANGES...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
2.5. COSETS AND LAGRANGE’S THEOREM 125 2.5.5. What is the analogue of Proposition 2.5.3 , with left cosets replaced with right cosets? 2.5.6. Let H be a subgroup of a group G . Show that aH 7! Ha ± 1 defines a bijection between left cosets of H in G and right cosets of H in G . (The index of a subgroup was defined in terms of left cosets, but this observation shows that we get the same notion using right cosets instead.) 2.5.7. For a subgroup N of a group G , prove that the following are equiv- alent: (a) N is normal. (b) Each left coset of N is also a right coset. That is, for each a 2 G , there is a b 2 G such that aN D Nb . (c) For each a 2 G , aN D Na . 2.5.8. Suppose N is a subgroup of a group G and ŒG W D 2 . Show that N is normal using the criterion of the previous exercise. 2.5.9. Show that if G is a finite group and N is a subgroup of index 2, then for elements a and b of G , the product ab is an element of N if, and only if, either both of a and b are in N or neither of a
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online