A maximal subgroup is a subgroup that is not contained in any larger proper subgroup of a given group, meaning it is maximal with respect to inclusion. Maximal subgroups play a crucial role in group theory as they help in understanding the structure of a group and its composition series.

Maximal Subgroup

Larger proper subgroup

Group theory is a branch of abstract algebra that studies the algebraic structures known as groups, which are sets equipped with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. It provides a unifying framework for understanding symmetry in mathematical objects and has applications across various fields including physics, chemistry, and computer science.

Group Theory

Structure of a group

A composition series is a finite sequence of subgroups in a group such that each is a normal subgroup of the previous one, and the quotient groups are simple. This series provides a way to analyze the structure of a group by breaking it down into simpler, well-understood components.

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