Elementary Properties of
Groups
4 Basic Properties
1.
2.
3.
4.
•
•
•
Uniqueness of the Identity
Cancellation
Uniqueness of Inverses
Shoes and Socks Property
What: Understand the property
Why: Prove the property
How: Use the property
1. Uniqueness of Identity
• In a group G, there is only one identity element.
• Proof: Suppose both e and e' are identities of G.
Then,
1. ae = a for all a in G, and
2. e'a = a for all a in G.
Let a = e' in (1) and a = e in (2).
Then (1) and (2) become
(1) e'e = e', and (2) e'e = e.
It follows that e = e'.
To use uniqueness of identity
• If ax = x for all x in some group G.
• Then a most be the identity in G!
*mod 40
5
15
25
35
5
25
35
5
15
15
35
25
15
5
25
35
5
15
15
5
25
35
35
25
Find e.
e = 25!
2. Cancellation
• In a group G, the right and left cancellation
laws hold. That is,
ba = ca implies b = c (right cancellation)
ab = ac implies b = c (left cancellation)
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 = ce by the definition of inverses
=> b = c by the definition of the identity.
Proof of left cancellation
• Similar.
• Put it in your proof notebook.
When not to use cancellation
• In D4
R90D = D'R90
• You cannot cross cancel, since D ≠ D'
• Order matters!
3. Uniqueness of inverses
• For each element a in a group G, there is a
unique element b in G such that ab=ba=e.
• Proof: Suppose b and c are both inverses
of a.
Then ab = e and ac = e
so ab = ac.
Cancel on the left to get b = c.
4. Shoes and Socks
• For group elements a and b,
•
(ab)-1 =b-1a-1
Proof:
(ab)(b-1a-1) = (a(bb-1))a-1
=(ae)a-1
= aa-1
=e
Since inverses are unique, b-1a-1 must be (ab)-1