Elementary Properties of Groups .ppt

properties of groups

1. Elementary Properties of
Groups

2. 4 Basic Properties
1.Uniqueness of the Identity
2.Cancellation
3.Uniqueness of Inverses
4.Shoes and Socks Property
•What: Understand the property
•Why: Prove the property
•How: Use the property

3. 1. Uniqueness of Identity
•In a group G, there is only one identity element.
•Proof: Suppose both eand e'are identities of G.
Then,
1. ae = afor all ain G, and
2. e'a = a for all ain G.
Let a= e'in (1) and a= ein (2).
Then (1) and (2) become
(1) e'e= e', and (2) e'e = e.
It follows that e= e'.

4. To use uniqueness of identity
•If ax= xfor all xin some group G.
•Then amost be the identity in G!
Find e.
e = 25!
*mod 40 5 15 25 35
5 25 35 5 15
15 35 25 15 5
25 5 15 25 35
35 15 5 35 25

5. 2. Cancellation
•In a group G, the right and left cancellation
laws hold. That is,
ba= caimplies b = c(right cancellation)
ab= acimplies b= c(left cancellation)

6. Proof: Right cancellation
•Let G be a group with identity element e.
Suppose ba=ca.
Let a'be an inverse of a. Then
(ba)a' = (ca)a'
=> b(aa') = c(aa') by associativity
=> be = ceby the definition of inverses
=> b = c by the definition of the identity.

7. Proof of left cancellation
•Similar.
•Put it in your proof notebook.

8. When not to use cancellation
•In D
4
R
90D = D'R
90
•You cannot cross cancel, since D ≠ D'
•Order matters!

9. 3. Uniqueness of inverses
•For each element ain a group G, there is a
unique element bin G such that ab=ba=e.
•Proof: Suppose band care both inverses
of a.
Then ab= eand ac= e
so ab = ac.
Cancel on the left to get b = c.

10. 4. Shoes and Socks
•For group elements aand b,
(ab)
-1
=b
-1
a
-1
•Proof:
(ab)(b
-1
a
-1
)=(a(bb
-1
))a
-1
=(ae)a
-1
=aa
-1
=e
Sinceinversesareunique,b
-1
a
-1
mustbe(ab)
-1