GATE 2024 Group Theory | Counting Group Homomorphisms & Normal Subgroups | CSIR-NET ,GATE , & TIFR

In this video, we discuss a very interesting and conceptually important Group Theory problem from GATE 2024 involving the number of group homomorphisms. This seemingly simple problem tests several important ideas of finite group theory, including: 🔹 Group homomorphisms 🔹 Kernel and image of a homomorphism 🔹 Elements of finite order 🔹 Cyclic groups 🔹 Symmetric group \(S_4\) 🔹 Normal subgroups 🔹 Counting techniques in Group Theory 🔹 Orders and cycle types of permutations The key to solving this problem is to understand the relationship between a homomorphism from a cyclic group and the possible images of its generator. This makes the problem particularly useful for developing conceptual problem-solving skills rather than relying on memorized formulas. 🎯 Highly relevant for: ✅ GATE Mathematics ✅ CSIR-NET Mathematical Sciences ✅ NBHM ✅ TIFR ✅ ISI Entrance ✅ IIT JAM Mathematics If you are preparing for competitive mathematics examinations, don't skip this problem! It combines multiple Group Theory concepts into a short but highly rewarding question. 📌 What you will learn How to count homomorphisms systematically How element orders control possible homomorphisms How permutation cycle types help in counting How normal subgroups arise naturally in homomorphism problems A useful strategy for tackling similar exam questions 👍 Like | Share | Subscribe to MathoPhilia for more GATE, CSIR-NET, NBHM, TIFR and advanced mathematics problems. 📚 Email : mathophilia2001@gmail.com and WhatsApp group link : https://chat.whatsapp.com/E1SbYjDnaj1... #GATE2024 #GroupTheory #GATEMathematics #GroupHomomorphism #NormalSubgroup #CSIRNET #NBHM #TIFR #ISI #Mathematics #GATEPreparation #MathoPhilia #AbstractAlgebra #PermutationGroup