Permutation Groups Practice Problems and Solutions

Struggling to apply abstract algebra theorems to actual symmetric and alternating groups? Transitioning from pure theory to solving concrete problems is one of the biggest hurdles in group theory. This text-based course bridges that gap by guiding you through structured problems and detailed, step-by-step written solutions. You will learn how to analyze cycle structures, compute products, and prove core algebraic properties with confidence. What you'll learn: Understand foundational definitions of symmetric and alternating groups; Practice decomposing permutations into disjoint cycles and transpositions; Calculate the order, sign, and inverse of various permutations; Apply Lagrange's theorem and conjugacy classes to permutation subgroups; Prove structural properties of permutation groups using clear mathematical writing. We begin with essential terminology, basic definitions, and foundational properties before moving into progressively challenging written exercises. This course is designed for undergraduate mathematics students, computer science learners, or self-taught enthusiasts who have a basic introduction to groups but need practical, solved problems to solidify their understanding. Grab your notebook, read through the explanations, and master permutation groups today.