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 Notation – additive and multiplicative Definition of subgroup Examples of subgroups Conditions for subgroup Cyclic subgroups Generators

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 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 a-n = (a-1 )n

Properties of Powers Powers satisfy the following rules, for all integers n, m: an am = an+m (an) m = an m (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 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 = q m + 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