LS11-3 - S UMMARY OF 11/3 L ECTURE We reviewed a couple of...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
S UMMARY OF 11/3 L ECTURE We reviewed a couple of notions. Most importantly, recall that the order of an element g of a group G is defined to be the order of the cyclic subgroup generated by that element: | g | = |h g i| . Another way to think about | g | is that it is the smallest positive integer n such that g n = 1 (i.e. g 6 = 1 ,g 2 6 = 1 ,g 3 6 = 1 ,...,g n - 1 6 = 1 , but g n = 1 ). It’s a good exercise to figure out why the order satisfies this property. (Also, note that g,g 2 ,...,g n are all distinct!) Next, we discussed direct products. Given two sets A and B we define a new set A × B , called the direct product of A and B , to be the collection of all ordered pairs of elements, the first from A , the second from B . In mathish, A × B = { ( a,b ) : a A,b B } . For example, R × Z is the set of all ordered pairs of the form ( x,n ) where x is any real number and n is any integer. Thus, (3 . 14159 , - 7) R × Z , but (3 , 2 . 1) 6∈ R × Z . We then concluded class by going over Cauchy’s theorem once more, this time completing the
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

LS11-3 - S UMMARY OF 11/3 L ECTURE We reviewed a couple of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online