Dr. Upasana's maths blog

 The concept of neighbourhood is fundamental in topology – it formalizes the intuitive idea of "points close to x." Definition A neighbourhood of point x in a topological space X is any set N containing an open set U that contains x. Formally : x ∈ U ⊆ N where U is open Think: "any set with some breathing room around x" Simple Examples in ℝ For point x = 3: (2, 4) is a neighbourhood ✓ [2, 4] is a neighbourhood ✓ (2.9, 3.1) is a neighbourhood ✓ {3} is NOT a neighbourhood ✗ Open Neighbourhood An open neighbourhood is a neighbourhood that is itself open. Example: (2, 4) is an open neighbourhood of 3, but [2, 4] is not. ε-Neighbourhood (Metric Spaces) N_ε(x) = {y | d(x, y) < ε} All points within distance ε from x. In ℝ : N₀.₅(3) = (2.5, 3.5) In ℝ² : N₁(0,0) = open disk of radius 1 Deleted Neighbourhood N (x) * = N \ {x} – the neighbourhood with x removed. Used for defining limits (what happens near x, but not at x). Key Properties If N is a...

 After teaching subgroups, students ask: "Are all subgroups created equal?" Not quite! Some subgroups are special – we call them normal subgroups, and they unlock one of group theory's most powerful concepts. What is a Normal Subgroup? A subgroup H of G is normal (written H ⊴ G) if: gH = Hg for all g ∈ G In other words: left cosets equal right cosets! Alternative definition : gHg⁻¹ = H for all g ∈ G This means H is closed under conjugation. Why "Normal"? Normal subgroups are "well-behaved" – they allow us to construct quotient groups, which are fundamental to understanding group structure. Simple Examples Example 1: Every Subgroup of an Abelian Group is Normal In an abelian group, gH = Hg automatically (since gh = hg). So in Z₆, every subgroup is normal: {0} {0, 3} {0, 2, 4} Z₆ Example 2: The Center is Always Normal Z(G) = {z ∈ G | zg = gz for all g ∈ G} Since center elements commute with everything, Z(G) ⊴ G always! Example 3: N...

 Permutation groups are where abstract algebra meets concrete action. Every time you shuffle cards or solve a Rubik's cube, you're working with permutations! What is a Permutation? A permutation of a set X is a bijection (one-to-one and onto function) from X to itself. Simply put: it's a way to rearrange the elements. Example : For {1, 2, 3}, one permutation is: 1 → 2 2 → 3 3 → 1 We write this as (123) in cycle notation. Cycle Notation Instead of listing where each element goes, we write cycles: (123) means: 1→2, 2→3, 3→1 (12)(34) means: 1↔2, 3↔4 (1) or just e is the identity (nothing moves) Example : The permutation that swaps 1 and 3 while fixing 2: (13) or equivalently (13)(2) , usually written as just (13) The Symmetric Group S_n S_n = the group of all permutations of {1, 2, 3, ..., n} Order : |S_n| = n! Examples : S₃ has 3! = 6 elements S₄ has 4! = 24 elements S₅ has 5! = 120 elements Elements of S₃ S₃ = {e, (12), (13), (23), (123), (...