Day 06Part 06- Discrete mathematics – Trick for finding of group theory of infinite numbers.

Day 06Part 06- Discrete mathematics – Trick for finding of group theory of infinite numbers.

Trick for Finding Group Theory of Infinite Numbers – Discrete Mathematics

Group Theory is a fundamental topic in Discrete Mathematics that deals with sets and operations satisfying specific properties. When working with infinite groups, understanding how to identify and analyze their structure efficiently is crucial.

 Quick Tricks for Identifying Groups in Infinite Sets

 Step 1: Verify the Group Properties

A set $G$ with a binary operation $\ast$ forms a group if it satisfies these four properties:

Closure: If $a,b\in G$, then $a\ast b\in G$.
Associativity: $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in G$.
Identity Element (e): There exists an element $e\in G$ such that $a\ast e=e\ast a=a$.
Inverse Element: Every element $a\in G$ has an inverse a−1a^{-1}a−1 such that a∗a−1=a−1∗a=ea * a^{-1} = a^{-1} * a = ea∗a−1=a−1∗a=e.

Step 2: Consider the Type of Infinite Set

Infinite groups can be classified into:
Additive Groups: (e.g., $(\mathbb{Z},+)$, $(\mathbb{Q},+)$, $(\mathbb{R},+)$)
Multiplicative Groups: (e.g., (R∗,⋅)(\mathbb{R}^*, \cdot)(R∗,⋅) where R∗\mathbb{R}^*R∗ is the set of nonzero real numbers)
Cyclic Infinite Groups: (Generated by one element, like $(\mathbb{Z},+)$)
Non-Abelian Infinite Groups: (e.g., Matrix Groups, such as $GL(n,\mathbb{R})$, the general linear group of invertible matrices)

 Step 3: Shortcut to Identify Group Structure

Trick 1: If a set is closed under addition/multiplication and has an identity, check inverses to confirm it’s a group.
Trick 2: If a set is generated by one element (like $\mathbb{Z}$ under addition), it’s a cyclic group.
Trick 3: For matrix groups, check if the determinant is nonzero for invertibility.
Trick 4: If the operation follows commutativity, it’s an Abelian group (easier to work with!).

 Example Problems

Example 1: Is $(\mathbb{Z},+)$ a group?
Yes! It satisfies all four properties:

• Closure  (sum of two integers is an integer)
• Associativity
• Identity  (0 is the identity)
• Inverse  (for every $x$, $-x$ exists)

Example 2: Is (R∗,⋅)(\mathbb{R}^*, \cdot)(R∗,⋅) a group?
Yes! It includes all nonzero real numbers under multiplication.

Example 3: Is $(\mathbb{N},+)$ a group?
No! Natural numbers do not have inverses under addition.

 Conclusion

Using these quick tricks, you can easily determine if an infinite set forms a group. Mastering these methods helps in solving advanced problems in algebra, cryptography, and theoretical computer science!

Day 06Part 06- Discrete mathematics – Trick for finding of group theory of infinite numbers.

The theory and practice of financial engineeri·ng

DISCRETE MATHEMATICS

Discrete Mathematics

document resume

Group Theory

Day 06 – Part 06: Discrete Mathematics

Trick to Understand Group Theory of Infinite Sets (for GATE/UGC NET/CS students)

What is Group Theory? (Quick Refresher)

A Group (G, *) is a set G with a binary operation * that satisfies:

1. Closure: ∀ a, b ∈ G, a * b ∈ G

2. Associativity: (a * b) * c = a * (b * c)

3. Identity: ∃ e ∈ G such that ∀ a ∈ G, a * e = a = e * a

4. Inverse: ∀ a ∈ G, ∃ a⁻¹ ∈ G such that a * a⁻¹ = e

Trick to Identify Groups with Infinite Sets

Infinite sets can form groups if they satisfy the 4 conditions above. Here’s how to quickly check:

Common Infinite Sets That Form Groups

Set	Operation	Group?	Why?
ℤ (integers)	Addition	Yes	Closure, inverse, identity 0
ℝ (non-zero real numbers)	Multiplication	Yes	Identity = 1, inverse = 1/x
ℝ (all real numbers)	Multiplication	No	0 has no inverse
ℕ (natural numbers)	Addition	No	No inverse for any n ≠ 0
ℚ{0} (non-zero rationals)	Multiplication	Yes	Group under ×
ℂ{0} (non-zero complex)	Multiplication	Yes	Complex multiplicative group

TRICK 1: Use this Group Filter

Ask these questions:

Is the set closed under the operation?
Is there an identity element in the set?
Does every element have an inverse in the same set?
Is the operation associative?

If all yes → It’s a group.
If any fail → Not a group.

TRICK 2: Infinite Groups – Don’t Worry About Size

A group can be:

• Finite (like {0, 1, 2} under mod addition)

• Infinite (like ℤ under addition)

Even infinite groups follow the same four rules!

Common Examples in Exams

Question Example	Is it a group?	Reason
(ℤ, +)	Yes	Additive group of integers
(ℤ, ×)	No	Inverse not always exists
(ℝ \ {0}, ×)	Yes	Multiplicative group
(ℕ, +)	No	No additive inverse in ℕ

Short Mnemonic: CIAI

Closure
Identity
Associativity
Inverse

If all are satisfied, it’s a group.

Summary:

• Not all infinite sets form groups

• Always test for the 4 group properties

• Multiplicative groups must exclude 0

• Sets like ℤ, ℝ{0}, and ℚ{0} under proper operations often form infinite groups

Would you like:

• A PDF trick sheet with examples?

• A practice worksheet with GATE/UGC-NET-style questions on group theory?

Let me know and I’ll prepare it for you!