Explore the foundational concepts of group theory in mathematics. Investigate the properties and structures of groups, including group operations, subgroups, and group homomorphisms. Analyze key theorems such as Lagrange’s theorem and the isomorphism theorems, and their applications in various mathematical contexts. Examine the classification of finite groups, focusing on symmetric groups and cyclic groups. Explore advanced topics such as group actions, Sylow theorems, and solvable groups. Utilize abstract reasoning, mathematical proofs, and problem-solving skills to deepen understanding of group theory concepts. Present your analysis in a rigorous and well-structured mathematical exposition, demonstrating mastery of group theory principles.
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