Abstract Algebra:
Groups:
Informally, a group is a set with an operation that acts like addition.
Defn: A group is a set G together with an operation + that satisfies
the following axioms:
1. (closure) ∀a, b ∈ G, a + b ∈ G
2. (associative) ∀a, b ∈ G, (a + b) + c = a + (b + c)
3. (identity) ∃0 ∈ Gs.t.∀a ∈ G, a + 0 = a = 0 + a
4. (inverse) ∀a ∈ G∃b ∈ Gs.t.a + b = 0 = b + a
Fact: 0 is unique and inverse of a is unique and written −a.
A group is abelian (commutative) if ∀a, b ∈ G, a + b = b + a.
In this class, the emphasis will be on abelian groups.
Note: for abelian groups, axioms 3 and 4 can be simplified.
Examples with ordinary addition (identify 0 and −a):
Z
R
Examples of abelian groups.
Z with ordinary addition
R with ordinary addition
The set of even integers: 2Z
Z2 = {0, 1} with binary addition:
1. closure: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0
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2. associativity (check case-by-case; later we will have a better
way)
3. identity: 0
4. inverses −0 = 0, and −1 = 1.
5. abelian: yes
Z3 = {0, 1, 2} with ternary addition, defined by addition table:
entry in row x and column y is x + y:
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
This is addition of integers modulo 3: remainder of a+b divided by
3.
1. closure: view table
2. associativity (check case-by-case; later we will have a better
way)
3. identity: 0
4. inverses −0 = 0, and −1 = 2 and −2 = 1.
5. abelian: yes (symmetry of array (matrix) across main diagonal)
Rigid motions of a square: G = {I, R1, R2, R3, FD , FA, FH , FV }
Ri: counterclockwise rotation by iπ/2
FD : reflection across main diagonal
FA: reflection across anti-diagonal
FH : reflection across horizontal
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FV : reflection across vertical
– with composition of mappings as addition.
1. Closure: build 8 × 8 table.
Examples:
– Ri ◦ Rj = Ri+j
(mod 4)
– FA ◦ FD = R2
– FH ◦ FV = R2
– FA ◦ R1 = FH
– R1 ◦ FA = FV
2. Associativity: true for all mappings
f ◦ (g ◦ h)(x) = f (g(h(x))) = (f ◦ g) ◦ h(x)
3. Identity: I
4. Inverses: the identity, R2 and the reflections are their own
inverses, and −R1 = R3, −R3 = R1 (usually written as R1−1 = R3
and R3−1 = R1).
5. Non-abelian: FA ◦ R1 6= R1 ◦ FA
Notation: An denotes the set of all words of length n with alphabet
A, e.g., {0, 1}n denotes the set of all binary words of length n.
Can be viewed as a set of strings, sequences or vectors.
Fact: If G is a group, then so is Gn (with entry-by-entry addition)
for x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn)
x + y = (x1 + y1, x2 + y2, . . . , xn + yn)
The identity is (0, 0, . . . , 0) and −x = (−x1, −x2, . . . , −xn).
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Examples:
R2 is the plane; R3 is 3-dimensional space.
Zn2 is the set of binary words of length n with with entry-by-entry
binary addition.
Defn: Let G be a group, with addition operation +. Let H be a
nonempty subset of G. We say that H is a subgroup of G if H is a
group, with addition +.
Theorem: H is a subgroup of a group G iff
i. ∀a, b ∈ H, a + b ∈ H
ii. ∀a ∈ H, −a ∈ H
Proof:
if: We have closure (1) and inverses (4). Associativity (2) is inherited from associativity on G. Given a ∈ H, by (ii), we have −a ∈ H,
and by (i), we have 0 = a + (−a) ∈ H. So, we have Identity (3).
only if: These are necessary conditions for H to inherit the groups
structure from G.
Example 1: G = Z and H = 2Z.
Example 2: G = R2 and H is the line y = x:
H = {(x, x) : x ∈ R}
Example 3: H = 3-repetition code is a subgroup of G = Z23:
+ 000 111
000 000 111
111 111 000
−000 = 000, −111 = 111
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Examples of sets and operations which are not groups:
– the set of odd integers (violates closure)
– the set of nonnegative integers (violates inverses)
– Z with subtraction (violates associativity)
(Z, −) is not a group because a − (b − c) 6= (a − b) − c.
Defn: Let H be a subgroup of a group G. A coset of H in G is a
set of the form:
a + H := {a + h : h ∈ H}
where a ∈ G.
Note: a coset of H is a subset of G;
Note: a subgroup is always a coset (H = 0 + H), but a coset is
not necessarily a subgroup.
Example 1: G = Z and H = 2Z. The cosets of H in G are H
(the even integers) and 1 + H, the odd integers:
even + H = H, odd + H = 1 + H
Example 2: G = Z23 and H = the 3-repetition code {000, 111},
the cosets are
H = (1, 1, 1) + H, (1, 0, 0) + H = (0, 1, 1) + H,
(0, 1, 0) + H = (1, 0, 1) + H, (0, 0, 1) + H = (1, 1, 0) + H
Example 3: G = R2 and H = the line y = x, i.e., {(x, x) : x ∈
R}.
For fixed (u, v) ∈ R2,
(u, v) + H = {(u + x, v + x : x ∈ R}
So, the cosets are the lines y = w + x, w ∈ R (i.e., the lines with
slope 1).
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Notation: for a, b ∈ G, a − b means a + (−b).
Lagrange’s Theorem: Let H be a subgroup of a group G.
1. The cosets of H in G form a partition of G, i.e.
(a) The union of the cosets of H is all of G
(b) Any pair of distinct cosets is disjoint
2. If H is finite, then all cosets of H have the same size, namely
|H|.
3. If G is finite, then the number of cosets is
|G|
|H| .
Picture:
Compare Theorem with examples 1,2,3.
Proof:
1a. For any a ∈ G, a = a + 0 ∈ a + H, since 0 ∈ H. Thus,
G ⊆ ∪a∈G a + H.
The reverse inclusion is clear. Thus, the union of the cosets of H is
all of G.
1b. It suffices to show that for all a, b ∈ G, if a + H intersects
b + H, then a + H = b + H. It then suffices only to show that
a+H ⊆b+H
because the reverse inclusion would follow by reversing the roles of a
and b.
If a + H intersects b + H, then there exist h, h0 ∈ H such that
a + h = b + h0
Thus, using associativity
a = a + 0 = a + (h − h) = (a + h) − h = (b + h0) − h = b + (h0 − h)
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Now, let h00 ∈ H. Then again using associativity,
a + h00 = (b + (h0 − h)) + h00 = b + ((h0 − h) + h00) ∈ b + H
since (h0 − h) + h00 ∈ H. So, a + H ⊆ b + H, as desired.
2. It suffices to show that for all a ∈ G, there is a bijection
(one-to-one and onto mapping) from H to a + H. Let
f : H → a + H,
f (h) = a + h
f is onto because any element of a + H is of the form a + h for
some h ∈ H.
To see that f is one-to-one, we must show that if f (h) = f (h0),
then h = h0.
If f (h) = f (h0), then a + h = a + h0 and so
h = −a + a + h = −a + a + h0 = h0
(note that we have used associativity again here)
c. By part b, all cosets have size |H|. By part a, they cover G
disjointly.
Example 4: G = Z52 and H = C3 (the code given on page 4 of the
text); there are 8 cosets. In HW, you will find these cosets.
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