Download Math 3121 Lecture 4 Sections 5

Math 3121
Abstract Algebra I
Lecture 4
Sections 5-6: Subgroups and Cyclic
Groups
HW due
• Hand in: pages 45-49: 2, 19, 24, 31, 35
• Questions on HW not to hand in:
Pages 45-49: 1, 3, 5, 21, 25
Section 5: Subgroups
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Notation – additive and multiplicative
Definition of subgroup
Examples of subgroups
Conditions for subgroup
Cyclic subgroups
Generators
Notation for Group Operations
• Notation for group operations
– Typically either multiplicative or additive
– With multiplicative notation
• The binary operation is indicated by juxtaposition (infix notation with
no symbol) and the result is called the product.
• The identity is indicated by “1” (sometimes by “e” or other letters).
• The inverse is indicated by “-1” superscript.
• The binary operation can be either noncommutative or commutative.
– With additive notation
• The binary operation is indicated by + (infix notation) and the result is
called the sum.
• The identity is indicated by 0.
• The inverse indicated by a negative prefix.
• The binary operation is almost always commutative (abelian).
Some Examples of Additive Notation
• ℤn – The integers modulo n, where n is a
positive integer. Here, addition is defined by
ordinary addition modulo n (remainder when
the sum is divided by n). Explain in class.
• In class look at tables for ℤ2 ,ℤ3 ℤ4.
• Note that ℤn is isomorphic to the nth roots of
unity in the complex plane with multiplicative
notation. Examples?
Some Examples of Multiplicative
Notation
• Real nxn invertible matrices form a group.
Multiplication is indicated by juxtaposition.
• Note an nxn matrix is invertible whenever its
determinant is nonzero.
Definition of Subgroup
• Definition: Let <G, *> be a group. A subgroup of <G, *> is a
group whose set is a subset H of G and whose binary
operation is the binary operation induced on H by *.
• Note: We have reverted to a formal notation. The symbols
for the binary operation for the group and its subgroups are
normally written with the same symbol or with the same
additive or multiplicative notation even though they are
technically different functions. We will continue to use the
same symbology for both operations.
• Conditions: Not any subset of G will do. H must be closed
under the binary operation of <G, *>, and it must be a
group under this operation.
More Terminology
• Improper subgroup: Any group is a subgroup
of itself (called the improper subgroup)
• Trivial subgroup: The set consisting of just the
identity of a group is a subgroup (called the
trivial subgroup.
Examples of Groups and their
Subgroups
• nℤ is a subgroup of ℤ.
• If a1, a2 ,… , an, are real numbers then the subset of ℝn
given by { (x1, x2, …, xn) in ℝn | a1 x1 + a2 x2 + … + an xn =
0} is a subgroup of ℝn . (Likewise for integers, rationals, and
complex numbers.)
• < ℤ,+> is a subgroup of < ℚ,+>, < ℝ,+>, and < ℂ,+>.
Note the usual inclusion maps.
• Look at subgroups of ℤ4. (This is isomorphic to {1, -1, i, -i}).
• Let H be the group of all nxn matrices with determinant
equal to 1 is a subgroup of the invertible nxn matrices.
• More examples in the book – continuous functions form a
subgroup of a larger additive group of functions, likewise
for differentiable functions.
Criterion for Subset to be a Subgroup
Theorem: A subset H of the set of a group G is a subgroup iff
all of the following hold:
1) H is closed under the binary operation of G.
2) The identity element of G is in H.
3) H is closed under inverses.
Proof: (Comment) This is fairly immediate by checking details.
See the book. The main point is that closure of the binary
operation allow it to induce a binary operation on H. One
subtlety is that associativity of the binary operation of G
implies the associativity of the induced binary operation.
Condition 2) is essential. Without it, H could be empty, so
not have an identity.
The Inclusion Map
• Remark: If H is a subset of a set G, then the
map that takes any element of H to itself as an
element of G is called the inclusion map.
Whenever <H, *> is a subgroup of <G, *>, then
the inclusion map of H in G satisfies the
homomorphism condition. Loosely speaking,
the inclusion map is a homomorphism from
the subset to the group G.
Powers
• If G is a group with multiplicative notation, a is an element of G, and
n is a positive integer, then an is defined in such a way that
1)
2)
3)
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9)
10)
11)
a0 = 1, the identity of G
a1 = a
a2 = a a
a3 = a a a
… (and so on)
an has n factors of a for n positive
a-1 is the inverse of a
a-2 = a-1 a-1 = (a-1 )2
a-3 = a-1 a-1 a-1 = (a-1 )3
… (and so on)
a-n = (a-1 )n
Properties of Powers
• Powers satisfy the following rules, for all integers
n, m:
1) an am = an+m
2) (an) m = an m
3) (an) -1 = a-n
• Note that 1) can be shown by considering various
cases. So can 2) which is needed later. Note that
3) is a special case of 2.
• Note that 1) makes the function f: ℤ  G given
by f(n) = an a homomorphism (stated more
formally on next slide).
Power Function
• Let G be a group with multiplicative notation, and let a be
an element of G. Then the function f: ℤ  G given by f(n)
= an is called the power function. It is a homomorphism.
• Note: With additive notation, powers are written
multiplicatively and the formula for f looks like f(n) = n a.
Then the homomorphism property is given by (n+m) a = n a
+ m a, which looks like a distributive law.
• Powers were introduced informally. They could be more
carefully defined inductively, but this is beyond the scope
of the course, for now at least. We can treat the existence
of the power function and the fact that it is a
homomorphism as fundamental.
The Principle of Mathematical
Induction
• Mathematical Induction is a standard way to
define and prove things about the natural
numbers and integers. One way to state it is that
if S is a nonempty set of positive integers that
contains 1 and for which n in S implies that n+1 is
in S, then S is equal to the entire set of positive
integers. An equivalent statement that we will
use is that any nonempty subset of nonnegative
integers has a least element. This property is
called well-ordering.
Cyclic Subgroups and Power Notation
Theorem: Let G be a group with multiplicative notation. For
each element a of G, the set
H = { an | n in ℤ}
is a subgroup of G and is the smallest subgroup of G that
contains a.
Proof: See the book. Sketch: H satisfies the three properties
for subgroups on the previous theorem. Thus is a
subgroup.
To prove any subgroup K of G that contains a must contain
H: Let S be the set of all positive n such that an is not in K.
If S is empty, we are done. If S is nonempty, then let m be
the smallest member of S. Then am = am-1a. But am-1and a
are both in K, thus am is also.
Section 6: Cyclic Groups
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Generators
Abelian
Division Algorithm
Counting arguments
Generators of Cyclic Groups
• Let G be a group with multiplicative notation.
By the previous section, each a in G
determines a subgroup of integral powers of
a. We say a generates this subgroup and
denote it by <a>.
Definition: A group is called cyclic if it is equal to
<a> for some a in the group.
Order of a Group
• The order of a group is the number of
elements it contains.
• The order of any member of a group is the
number of elements in its cyclic group.
Cyclic Implies Abelian
Theorem: Every Cyclic group is abelian
(commutative).
Proof: If x and y are in the group, then there is
an a in the group and integers n, m such that x
= an and y = am. Then x y = an am = an+m = am+n
= am an = y x.
Division Algorithm
Theorem (Division Algorithm): If m is a positive integer
and n is any integer, then there exist unique integers q
and r such that
n=qm+r
and 0 ≤ r < m
Proof: This follows well-ordering (or by induction). Let’s
use well-ordering. Let S be the set of all nonnegative
integers of the form n – q m. S is nonempty: If n is
positive, let q=0, then n – m 0 is in S and it positive. If n
is negative, let q=n, then n – m (n) = -n(-1+m), which is
nonnegative. Let r be the least element of S. Then 0 ≤
r. If r ≥ m, then r = n – q m, for some q, so r-m = n –
(q+1) m is still in S, and thus not minimal. Thus r < m.
So 0 ≤ r < m. But r = n – q m implies that n = q m + r.
A subgroup of a Cyclic Group is Cyclic
Theorem: A subgroup of a cyclic group is cyclic.
Proof: In class
Subgroups of Integers Are Cyclic
Corollary: A subgroup of the integers is cyclic.
Proof: The integers are cyclic, hence any
subgroup is cyclic.
GCD
Definition (normal definition): The greatest
common divisor of two integers is the largest
integer that divides both.
Theorem: If n and m are integers, then there are
integers a and b such that gcd(n, m) = a n + b
m.
Proof: In class.
Examples
• Find gcd of 60 and 56
Relatively Prime
Definition: Two integers are relatively prime if
their gcd is 1.
Theorem
Theorem: A cyclic group is isomorphic to ℤ or to
ℤn. In the first case it is of infinite order, and
in the second case it is of finite order.
Proof: In class – use the power function. Note
that ax = ay iff x = y mod n, where n is the
order of the group if the group is finite. Two
cases: 1) an is never equal to e, and 2) an is
equal to e for some n.
Theorem
Theorem: Let G be a cyclic group of finite order
n and generated by a. Let b = as, for some
integer s. Then b generates a cyclic group of
order n/gcd(n, s). Furthermore, <as> = <at> iff
gcd(s, n) = gcd(t, n).
Proof: See book. Use ℤn as a model.
HW
• Don’t hand in:
Pages 66-67: 3, 5, 11, 17, 23, 45