Mathematics - Discrete Mathematics - Group-like Structures (part 1)

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Introduction

Hey it's a me again @drifter1!

Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Group-like Structures. The topic will be split into multiple parts.

So, without further ado, let's get straight into it!

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Subgroups

A subgroup H is a subset of a given group G, which still satisfies the four group requirements and is by itself also a group under the binary (or group) operation of G. For any such subgroup, the group order must be a divisor of the original group.

Proper Subgroups

If a subgroup H doesn't include the entire group G (i.e. H ⊂ G) then the subgroup is considered a proper subgroup.

Cyclic Subgroups

In a similar manner to groups, subgroups can also be cyclic, if each element can be written in the form xⁿ for some integer n and generator x.

Normal Subgroups

A subgroup H of group G is a normal subgroup of G if x h x^-1 ∈ H for all h ∈ H and x ∈ G.

In the case of an abelian group G, every subgroup H is a normal subgroup in G.

Cosets

For a subgroup H of a group G, two cosets (left and right) can be defined. The left coset of H is subset of G for which the elements can be expressed as xh, for any x ∈ G and h ∈ H. Similarly, the right coset is expressed as hx.

For an additive (+) group operation the cosets can be defined as x + h and h + x respectively.

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Semigroups

The requirements of groups are quite strict. That's the reason why additional algebraic structures such as semi-groups are defined, where some of the properties are eliminated.

For example. a semigroup (A, *) is a set A together with an binary operation * on A, which only satisfies closure and associativity.

Subsemigroups

Similar to groups, semigroups can also have sub-structures, known as subsemigroups. For a given semigroup A, a subsemigroup B is a semigroup which consists of a subset of A, or B ⊆ A. If B ⊂ A, then B is a proper subsemigroup of A.

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Monoids

Eliminating only the inverse property from groups, yields an algebraic structure which is known as a monoid. As such, a monoid satisfies closure, associativity and identity.

Submonoid

For any monoid M, it's possible to take a submonoid S, which consists of a subset of the original monoid, or S ⊆ M.

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RESOURCES:

References

1. https://www.javatpoint.com/discrete-mathematics-tutorial
2. https://mathworld.wolfram.com/Group.html
3. https://en.wikipedia.org/wiki/Group_(mathematics)

Images

1. 

Mathematical equations used in this article, have been generated using quicklatex.

Block diagrams and other visualizations were made using draw.io.

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Previous articles of the series

• Introduction → Discrete Mathematics, Why Discrete Math, Series Outline
• Sets → Set Theory, Sets (Representation, Common Notations, Cardinality, Types)
• Set Operations → Venn Diagrams, Set Operations, Properties and Laws
• Sets and Relations → Cartesian Product of Sets, Relation and Function Terminology (Domain, Co-Domain and Range, Types and Properties)
• Relation Closures → Relation Closures (Reflexive, Symmetric, Transitive), Full-On Example
• Equivalence Relations → Equivalence Relations (Properties, Equivalent Elements, Equivalence Classes, Partitions)
• Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
• Combinatorial Principles → Combinatorics, Basic Counting Principles (Additive, Multiplicative), Inclusion-Exclusion Principle (PIE)
• Combinations and Permutations → Factorial, Binomial Coefficient, Combination and Permutation (with / out repetition)
• Combinatorics Topics → Pigeonhole Principle, Pascal's Triangle and Binomial Theorem, Counting Derangements
• Propositions and Connectives → Propositional Logic, Propositions, Connectives (∧, ∨, →, ↔ and ¬)
• Implication and Equivalence Statements → Truth Tables, Implication, Equivalence, Propositional Algebra
• Proof Strategies (part 1) → Proofs, Direct Proof, Proof by Contrapositive, Proof by Contradiction
• Proof Strategies (part 2) → Proof by Cases, Proof by Counter-Example, Mathematical Induction
• Sequences and Recurrence Relations → Sequences (Terms, Definition, Arithmetic, Geometric), Recurrence Relations
• Probability → Probability Theory, Probability, Theorems, Example
• Conditional Probability → Conditional Probability, Law of Total Probability, Bayes' Theorem, Full-On Example
• Graphs → Graph Theory, Graphs (Vertices, Types, Handshake Lemma)
• Graphs 2 → Graph Representation (Adjacency Matrix and Lists), Graph Types and Properties (Isomorphic, Subgraphs, Bipartite, Regular, Planar)
• Paths and Circuits → Paths, Circuits, Euler, Hamilton
• Trees → Trees (Rooted, General and Binary), Tree Traversal, Spanning Trees
• Common Graph Problems → Shortest Path Problem, Graph Connectivity, Travelling Salesman Problem, Minimum Spanning Tree, Maximum Network Flow, Graph Coloring
• Binary Operations → Binary Operations (n-ary, Table Representation), Properties
• Groups → Groups (Properties, Theorems, Finite and Infinite, Abelian, Cyclic, Product, Homo-, Iso- and Auto-morphism)

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Final words | Next up

And this is actually it for today's post!

Next time we will continue on with more Group-like algebraic structures of Group Theory...

See ya!

Keep on drifting!