Math 320

Fall 04

Is it a Group?

 

            Note that addition and multiplication are associative (including in various mods), and that function composition is always associative (why?).

 

Set

Operation

Closed?

Associative?

Identity?

Inverses?

Is it a Group?

Natural Numbers

 

+

 

 

 

 

 

 

-

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

Integers  

 

+

 

 

 

 

 

 

-

 

 

 

 

 

 

x

 

 

 

 

 

Rationals  

 

+

 

 

 

 

 

 

-

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

Integers mod 3,  

+

 

 

 

 

 

 

+

 

 

 

 

 

 

+

 

 

 

 

 

 

+

 

 

 

 

 

 

x

 

 

 

 

 

 

x

 

 

 

 

 

 

x

 

 

 

 

 

 

x

 

 

 

 

 

U10 = units, mod 10,  (rel. prime to 10)

 

+

 

 

 

 

 

U10

x

 

 

 

 

 

U3

x

 

 

 

 

 

U5

x

 

 

 

 

 

U6

x

 

 

 

 

 

D3 = symmetries triangle

composition

 

 

 

 

 

C3= rotations of triangle

composition

 

 

 

 

 

Copyright 2005, Debra K. Borkovitz.  You may copy or edit this material for non-profit, educational use only.

 

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