This section introduces the basics of group theory, which is crucial for counting arrangements of objects, especially when symmetrical arrangements are considered identical. The chapter assumes readers have a basic understanding of group theory, including the definitions of subgroups, the order of a group, normal subgroups, homomorphisms, and cosets. - **Subgroup**: A subset \( H \) of a group \( G \) is a subgroup if it forms a group under the same operation as \( G \). - **Order of a Group**: The order of a finite group \( G \) is the number of its elements. A subgroup generated by an element \( g \) is finite if \( g \) has finite order. - **Normal Subgroup**: A subgroup \( H \) of \( G \) is normal if \( gH = Hg \) for all \( g \in G \). - **Homomorphism**: A homomorphism from \( G \) to \( H \) is a function \( \phi \) such that \( \phi(xy) = \phi(x)\phi(y) \) for all \( x, y \in G \). The kernel of \( \phi \) is the set \( \{x \in G : \phi(x) = 1\} \). - **Cosets**: For a subgroup \( H \) of \( G \), the right coset \( Hx \) is \( \{hx : h \in H\} \) and the left coset \( xH \) is \( \{xh : h \in H\} \). The section also includes Lagrange's Theorem, which states that the order of a subgroup divides the order of the group, and exercises to reinforce the concepts.This section introduces the basics of group theory, which is crucial for counting arrangements of objects, especially when symmetrical arrangements are considered identical. The chapter assumes readers have a basic understanding of group theory, including the definitions of subgroups, the order of a group, normal subgroups, homomorphisms, and cosets. - **Subgroup**: A subset \( H \) of a group \( G \) is a subgroup if it forms a group under the same operation as \( G \). - **Order of a Group**: The order of a finite group \( G \) is the number of its elements. A subgroup generated by an element \( g \) is finite if \( g \) has finite order. - **Normal Subgroup**: A subgroup \( H \) of \( G \) is normal if \( gH = Hg \) for all \( g \in G \). - **Homomorphism**: A homomorphism from \( G \) to \( H \) is a function \( \phi \) such that \( \phi(xy) = \phi(x)\phi(y) \) for all \( x, y \in G \). The kernel of \( \phi \) is the set \( \{x \in G : \phi(x) = 1\} \). - **Cosets**: For a subgroup \( H \) of \( G \), the right coset \( Hx \) is \( \{hx : h \in H\} \) and the left coset \( xH \) is \( \{xh : h \in H\} \). The section also includes Lagrange's Theorem, which states that the order of a subgroup divides the order of the group, and exercises to reinforce the concepts.