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Modern Algebra

by: Otilia Murray I

21

0

1

Modern Algebra MAT 150A

Otilia Murray I
UCD
GPA 3.88

Monica Vazirani

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COURSE
PROF.
Monica Vazirani
TYPE
Class Notes
PAGES
1
WORDS
KARMA
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Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 150A at University of California - Davis taught by Monica Vazirani in Fall. Since its upload, it has received 21 views. For similar materials see /class/187385/mat-150a-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15
MAT 150A 2Miscellaneous10 CONNOR SEMPEK 10 November 2008 Let HK be subgroups of a group G and g 6 Ci The set HgKzEGlzhgk forsomehEHkEK is called a double coset a Prove that double cosets partition Gr Proof Note that H and K are both nonempty so it follows that HgK is nonempty for all g 6 G Let gg E G and suppose that HgK HgK 0 Then 3 I such that z E HgK and z E HgKi Then I hgk h g k for some hh E Hkk E Ki So hgk hgk 9 hilhgkkil HgK HhilhgkkilK HgKi Hence any two double cosets are either equal or disjointi Observe that V h E Hg E G 3 g E G such that g hgi Moreover every double coset can be rewritten as HgK U thi hEH Thus by properties of cosets U HgK U th U g KG 960 yea yea heH b Do all double cosets have the same order No consider the subgroups 6 13 and 6 23 of 53 then 67 1312 7 23 12 123 67 1313 7 23 67 13 23 132 Hence H67 131267 23l 2 i 4 H67 131367 23l

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