LS9-10 - S UMMARY OF 9/10 L ECTURE Last time we discussed...

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: S UMMARY OF 9/10 L ECTURE Last time we discussed associativity as though it were a special property, but when does it fail? Here’s one example: what does 4 ÷ 2 ÷ 2 mean? Well, it could mean (4 ÷ 2) ÷ 2 = 1 , or it could mean 4 ÷ (2 ÷ 2) = 4 . The point is, the expression 4 ÷ 2 ÷ 2 is ambiguous; so, for example, ( Q × , ÷ ) is not a group, because associativity fails to hold. (There are other reasons it’s not a group, as well.) Next, we proved that any group has a unique identity element (hence, we talk about the identity) and that every element has a unique inverse. Of our three examples of groups so far, recall that example (v) was a bit different from the other two: f ◦ g 6 = g ◦ f in general. However, for some choices of f and g the two expressions are equal; for example, 1 commutes with every element f ∈ A ( S ) . More generally, given a group ( G, @ ) , we say two elements a,b ∈ G commute if a @ b = b @ a . If every element of G commutes with every other element, then we call...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

LS9-10 - S UMMARY OF 9/10 L ECTURE Last time we discussed...

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