Getting started with Group Theory

Getting started with Group Theory

I can't finish writing this thing at half past one...

Group

Define $ G = {a, B, c, \ ldots }$, $ * $ for binary operations on the set $ G $

When the set $ G $ meets the following nature under the operation $ * $, we call the set $ G $ a group under the operation $ * $, for short, $ G $ is a group.

• Closed: $ \ forall a, B \ in G, \ exists c \ in G, a ^ {\ ast} B = c $
• Combination law: $ \ forall a, B, c \ in G, (a * B) * c = a * (B * c) $
• Unit: $ \ forall a \ in G, \ exists e \ in G, a * e = e * a = a $
• Reverse meta: $ \ forall a \ in G, \ exists B \ in G, a * B = B * a = e $, note $ B = a ^ {-1} $

$ * $ Can be any operation, for example, $ +-\ times/$

Multiplication group: Multiplication Group

Addition group: The operation rule is the addition group.

Finite Group: the elements in a group are finite groups.

Wireless group: the elements in a group are infinite groups.

Price of a finite group: number of elements in a finite group

 

Replacement

Trap ....