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The chapter introduces the concept of subgroups within the context of group theory, outlining the definitions, properties, and important theorems such as Lagrange's theorem. It details methods to determine whether a subset is a subgroup, explores cyclic subgroups, and discusses the significance of left and right cosets in relation to subgroup equivalency. Furthermore, the chapter highlights the implications of Lagrange's theorem for finite groups, including relationships between the orders of subgroups and their parent groups.
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ch64.pdfClass Notes
Memorization
What we have learnt
- A subgroup, formed from a s...
- Lagrange's theorem states t...
- Left and right cosets help ...
Final Test
Revision Tests
What we have learnt
- A subgroup, formed from a subset of a group, must satisfy specific group axioms including closure and the presence of inverses.
- Lagrange's theorem states that the order of a subgroup divides the order of the finite parent group.
- Left and right cosets help in examining the structure of groups and can reveal properties about subgroup equivalences.
Key Concepts
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Term: Subgroup
Definition: A subgroup is a subset of a group that is itself a group under the operation defined on the parent group.
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Term: Lagrange's Theorem
Definition: A theorem stating that for a finite group, the order of a subgroup divides the order of the parent group.
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Term: Cosets
Definition: The left coset of a subgroup is formed by multiplying a group element by each element of the subgroup, while the right coset multiplies elements of the subgroup by the group element.