Finite group theory studies groups with finitely many elements, including their subgroups, normal subgroups, group actions, and representations. Finiteness turns structural questions into arithmetic restrictions on group order.
For example, Lagrange's theorem says a subgroup order divides the group order, while the Sylow theorems locate subgroups of prime-power order and constrain their number. The symmetric groups encode permutations, and the dihedral groups encode polygon symmetries. These examples connect finite algebra to counting and geometry; finite simple groups serve as the possible factors of composition series.
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