1) What is a Group? - The Core Definition 🛡️
Welcome to the fascinating world of Group Theory! Groups are fundamental algebraic structures that appear throughout mathematics, physics, computer science, and beyond. They abstract the idea of symmetry and operations. In essence, a group is a set equipped with an operation that combines any two elements to form a third element, in a way that satisfies certain rules.
A Group is a set \(G\) together with a binary operation \( * \) (often called "multiplication", but can be addition or other operations in different contexts) that satisfies the following four axioms:
Group Axioms
- Closure: For all \( a, b \in G \), \( a * b \in G \). (The result of combining any two elements is still in the set.)
- Associativity: For all \( a, b, c \in G \), \( (a * b) * c = a * (b * c) \). (The way you group operations doesn't change the result.)
- Identity Element: There exists an element \( e \in G \) such that for all \( a \in G \), \( a * e = e * a = a \). (There's a special "do-nothing" element.)
- Inverse Element: For every \( a \in G \), there exists an element \( a^{-1} \in G \) such that \( a * a^{-1} = a^{-1} * a = e \). (Every element has an "undo" element or inverse.)
If a set \(G\) with an operation \( * \) satisfies all four axioms, then \( (G, *) \) is a group. Sometimes we simply say "G is a group" when the operation is clear from the context.
Example 1: Integers under Addition \( (\mathbb{Z}, +) \) - An Abelian Group
Let \( \mathbb{Z} \) be the set of integers \( \{..., -2, -1, 0, 1, 2, ...\} \) and the operation be ordinary addition \( + \). Let's check the group axioms:
- Closure: If \( a, b \) are integers, then \( a + b \) is also an integer. (True)
- Associativity: For integers \( a, b, c \), \( (a + b) + c = a + (b + c) \). (True)
- Identity Element: The identity element is \( 0 \), since for any integer \( a \), \( a + 0 = 0 + a = a \). (True, identity element is 0)
- Inverse Element: For every integer \( a \), the inverse is \( -a \), since \( a + (-a) = (-a) + a = 0 \) (the identity element). (True, inverse of \(a\) is \(-a\))
A group \( (G, *) \) is called Abelian (or Commutative) if for all \( a, b \in G \), \( a * b = b * a \). If the operation is not always commutative, the group is called non-Abelian.
Example 2: Non-zero Real Numbers under Multiplication \( (\mathbb{R} \setminus \{0\}, \cdot) \) - Another Abelian Group
Let \( \mathbb{R} \setminus \{0\} \) be the set of all real numbers except zero, and the operation be ordinary multiplication \( \cdot \).
- Closure: If \( a, b \) are non-zero real numbers, then \( a \cdot b \) is also a non-zero real number. (True)
- Associativity: For real numbers \( a, b, c \), \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \). (True)
- Identity Element: The identity element is \( 1 \), since for any non-zero real number \( a \), \( a \cdot 1 = 1 \cdot a = a \). (True, identity is 1)
- Inverse Element: For every non-zero real number \( a \), the inverse is \( \frac{1}{a} \), since \( a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = 1 \) (the identity element). (True, inverse of \(a\) is \(1/a\))
Example 3: Integers under Multiplication \( (\mathbb{Z}, \cdot) \) - NOT a Group
Consider the set of integers \( \mathbb{Z} \) with multiplication \( \cdot \). Let's check axioms:
- Closure: Yes, product of two integers is an integer.
- Associativity: Yes, multiplication of integers is associative.
- Identity Element: Yes, the identity element is \( 1 \), since \( a \cdot 1 = 1 \cdot a = a \) for any integer \( a \).
- Inverse Element: Does every integer have a multiplicative inverse in \( \mathbb{Z} \)? No. For example, the inverse of \( 2 \) would need to be \( \frac{1}{2} \), but \( \frac{1}{2} \) is not an integer. Only \( 1 \) and \( -1 \) have integer inverses in \( \mathbb{Z} \). (Axiom 4 fails!)
2) Basic Properties of Groups - Consequences of Axioms 💡
From just the four group axioms, we can deduce several important properties that hold for all groups. These properties simplify working with groups and are fundamental in group theory.
2.1 Uniqueness of Identity and Inverses
In any group \( (G, *) \):
- The Identity Element is Unique: There is only one element \( e \in G \) that satisfies the identity axiom.
- Inverses are Unique: For each \( a \in G \), there is exactly one inverse element \( a^{-1} \in G \).
(Proofs of uniqueness can be shown using group axioms, but are omitted here for brevity at this introductory level. They are typically shown by assuming there are two and showing they must be the same).
2.2 Cancellation Laws
In any group \( (G, *) \), the cancellation laws hold:
- Left Cancellation: If \( a * b = a * c \), then \( b = c \) for all \( a, b, c \in G \).
- Right Cancellation: If \( b * a = c * a \), then \( b = c \) for all \( a, b, c \in G \).
(These can be proven using the existence of inverses. For example, for left cancellation, multiply both sides of \( a * b = a * c \) on the left by \( a^{-1} \)).
2.3 Socks-Shoes Theorem (Inverse of a Product)
For any elements \( a, b \) in a group \( (G, *) \), the inverse of the product \( (a * b) \) is given by:
\( (a * b)^{-1} = b^{-1} * a^{-1} \)
(Note the order reversal – like putting on and taking off socks and shoes!).(Proof involves checking that \( (a*b) * (b^{-1} * a^{-1}) = e \) and \( (b^{-1} * a^{-1}) * (a*b) = e \) using associativity and inverse properties).
3) Subgroups - Groups within Groups 🔗
Just like subspaces within vector spaces, groups can contain "subgroups". A subgroup is a subset of a group that is itself a group under the same operation.
A Subgroup \(H\) of a group \( (G, *) \) is a subset \( H \subseteq G \) such that \(H\) itself is a group under the same operation \( * \) restricted to \(H\).
3.1 Subgroup Tests - How to check if a subset is a subgroup
To check if a subset \(H\) of a group \( (G, *) \) is a subgroup, we don't need to verify all four group axioms for \(H\) from scratch. We can use shorter subgroup tests:
Subgroup Tests
Let \( (G, *) \) be a group and \( H \subseteq G \) be a non-empty subset of \(G\). Then \(H\) is a subgroup of \(G\) if either of the following conditions hold:
- Two-Step Subgroup Test:
- (a) Closure: For all \( a, b \in H \), \( a * b \in H \).
- (b) Inverses: For all \( a \in H \), \( a^{-1} \in H \).
- One-Step Subgroup Test (for non-empty \(H\)):
- For all \( a, b \in H \), \( a * b^{-1} \in H \).
Note: For finite groups, the closure condition alone is sufficient to guarantee that a non-empty subset is a subgroup.
Example 4: Subgroup of \( (\mathbb{Z}, +) \) - Even Integers
Let \( 2\mathbb{Z} = \{ ..., -4, -2, 0, 2, 4, ... \} \) be the set of even integers. Is \( (2\mathbb{Z}, +) \) a subgroup of \( (\mathbb{Z}, +) \)?
Using Two-Step Subgroup Test:
- (a) Closure: Let \( a, b \in 2\mathbb{Z} \). Then \( a = 2m \) and \( b = 2n \) for some integers \( m, n \). \( a + b = 2m + 2n = 2(m + n) \). Since \( m + n \) is an integer, \( a + b \) is an even integer, so \( a + b \in 2\mathbb{Z} \). (Closure holds)
- (b) Inverses: Let \( a \in 2\mathbb{Z} \). Then \( a = 2m \) for some integer \( m \). The inverse of \( a \) in \( (\mathbb{Z}, +) \) is \( -a = -2m = 2(-m) \). Since \( -m \) is an integer, \( -a \) is an even integer, so \( -a \in 2\mathbb{Z} \). (Inverses exist in \(2\mathbb{Z}\))
Example 5: Subset that is NOT a Subgroup - Positive Integers in \( (\mathbb{Z}, +) \)
Let \( \mathbb{Z}^+ = \{ 1, 2, 3, ... \} \) be the set of positive integers. Is \( (\mathbb{Z}^+, +) \) a subgroup of \( (\mathbb{Z}, +) \)?
No. Although \( (\mathbb{Z}^+, +) \) is closed under addition (sum of two positive integers is positive), it does not contain the identity element \( 0 \) of \( (\mathbb{Z}, +) \), and positive integers do not have inverses within \( \mathbb{Z}^+ \) under addition (e.g., inverse of 2 is -2, which is not in \( \mathbb{Z}^+\) ). Therefore, \( (\mathbb{Z}^+, +) \) is not a subgroup of \( (\mathbb{Z}, +) \).
4) Cyclic Groups - Groups Generated by One Element 💫
Cyclic groups are among the simplest and most fundamental types of groups. They are generated by a single element, meaning all elements in the group can be obtained by repeatedly applying the group operation to this one element (and its inverse).
A group \( (G, *) \) is called Cyclic if there exists an element \( g \in G \) such that every element of \(G\) can be written as \( g^n \) for some integer \( n \). Such an element \( g \) is called a generator of \(G\), and we write \( G = \langle g \rangle = \{ ..., g^{-2}, g^{-1}, g^0=e, g^1=g, g^2, ... \} \). Here, \( g^n \) means \( g * g * ... * g \) (\(n\) times) if \( n > 0 \), \( g^0 = e \) (identity), and \( g^n = (g^{-1})^{-n} \) if \( n < 0 \). If operation is addition, \( ng \) means \( g + g + ... + g \) (\(n\) times).
Example 6: Cyclic Group \( (\mathbb{Z}, +) \)
Is \( (\mathbb{Z}, +) \) a cyclic group?
Yes. Consider the element \( 1 \in \mathbb{Z} \). We can generate any integer by repeatedly adding \( 1 \) or \( -1 \).
- \( 2 = 1 + 1 \) ( \( 1^2 \) in additive notation, meaning 2 times operation with generator 1)
- \( 3 = 1 + 1 + 1 \) ( \( 1^3 \))
- \( 0 \) is the identity ( \( 1^0 \))
- \( -1 \) is the inverse of \( 1 \) ( \( 1^{-1} \))
- \( -2 = (-1) + (-1) \) ( \( 1^{-2} \))
- and so on...
Example 7: Cyclic Group of Integers Modulo \(n\), \( (\mathbb{Z}_n, +_n) \)
Consider \( \mathbb{Z}_4 = \{ 0, 1, 2, 3 \} \) under addition modulo 4, \( +_4 \). Is \( (\mathbb{Z}_4, +_4) \) cyclic?
Let's check if \( 1 \) is a generator:
- \( 1^1 = 1 \)
- \( 1^2 = 1 +_4 1 = 2 \)
- \( 1^3 = 1 +_4 1 +_4 1 = 3 \)
- \( 1^4 = 1 +_4 1 +_4 1 +_4 1 = 4 \equiv 0 \pmod{4} \) (identity element in \( \mathbb{Z}_4 \) is 0)
- \( 1^5 = 1^4 +_4 1 = 0 +_4 1 = 1 \) (repeats)
5) Practice Questions 🎯
5.1 Fundamental – Build Skills
1. Determine if the set of integers with operation subtraction \( (\mathbb{Z}, -) \) is a group. Check all four group axioms.
2. Determine if the set of positive real numbers under multiplication \( (\mathbb{R}^+, \cdot) \) is a group. Check all four group axioms.
3. Is the set of integers modulo 5 under multiplication \( (\mathbb{Z}_5, \cdot_5) \) a group? Why or why not? (Consider zero element).
4. Is the set of non-zero integers modulo 5 under multiplication \( (\mathbb{Z}_5 \setminus \{0\}, \cdot_5) \) a group? Check all four group axioms.
5. Let \( G \) be a group. Show that the identity element \( e \) is its own inverse (i.e., \( e^{-1} = e \)).
6. Let \( G \) be an Abelian group. Show that for any \( a, b \in G \), \( (a * b)^{-1} = a^{-1} * b^{-1} \). (Compare to Socks-Shoes theorem for general groups).
7. Is the set of odd integers a subgroup of \( (\mathbb{Z}, +) \)? Justify your answer using subgroup tests.
8. Is the set of all integer multiples of 3, denoted \( 3\mathbb{Z} = \{ ..., -6, -3, 0, 3, 6, ... \} \), a subgroup of \( (\mathbb{Z}, +) \)? Justify.
9. Is \( (\mathbb{Z}_6, +_6) \) a cyclic group? Find all generators of \( (\mathbb{Z}_6, +_6) \) if it is cyclic.
10. Is the group of non-zero integers modulo 5 under multiplication \( (\mathbb{Z}_5 \setminus \{0\}, \cdot_5) \) cyclic? Find a generator if it is.
5.2 Challenging – Push Limits 💪🚀
1. Let \( G \) be a group. Prove the left cancellation law: If \( a * b = a * c \), then \( b = c \). (Use group axioms).
2. Prove that in any group \( (G, *) \), the identity element is unique.
3. Prove that in any group \( (G, *) \), the inverse of each element is unique.
4. Let \( G \) be a group and \( a \in G \). Define \( \langle a \rangle = \{ a^n \mid n \in \mathbb{Z} \} \) (cyclic subgroup generated by \(a\)). Show that \( \langle a \rangle \) is indeed a subgroup of \( G \). It is the "smallest" subgroup containing \(a\).
5. (Conceptual) Explain in your own words why the concept of a "group" is so fundamental in mathematics. Think about the axioms and what kind of structures they describe. Where might you expect to find groups in different areas of math and science?
6) Summary 🎉
- Group Definition: A set with a binary operation satisfying Closure, Associativity, Identity, and Inverse axioms.
- Abelian Group: A group where the operation is commutative.
- Group Properties: Unique identity, unique inverses, cancellation laws, Socks-Shoes Theorem for inverses of products.
- Subgroup: A subset of a group that is itself a group under the same operation. Subgroup tests provide efficient ways to check if a subset is a subgroup.
- Cyclic Group: A group generated by a single element. All elements are powers of a generator. Examples: \( (\mathbb{Z}, +) \), \( (\mathbb{Z}_n, +_n) \).
Congratulations on starting your journey into Group Theory! You've now learned the fundamental definition of a group and explored key examples and properties. Understanding these foundations is crucial as we move to more advanced group concepts. Keep practicing with examples and the axioms – you're building a solid base in abstract algebra! 🛡️🔑
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