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Abstract Algebra (Group (Homomorphism
Let \(G, G'\) be two groups; then…
Abstract Algebra
Group
Definitions
- Operation under closure
- Associative
- Include identity element (unique identity in fact)
- For all element, there exists inverse element (unique element in fact)
Subgroup
Just verify that subgroup H in group G:
- Closure in H -> associative
- Inverse element exists in H -> identity exists
Trivial Subgroups & Proper Subgroups
Trivial:
\(G, \{e\}\)
Proper: otherwise subgroups
Cyclic Subgroup of \(G\) generated by \(a\) is \(\{a^i | i \in \mathbb{N} \} \), denoted \((a)\)
Notation
\(G=(S, *)\)
G: set
*: operation
Theorems & Lemmas
Problem 2.2.2 [ref]
G has properties:
- Finite set
- Closed under operation
- Cancellation
Then G is a group.
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Homomorphism
Let \(G, G'\) be two groups; then the mapping \(\psi:G \rightarrow G'\) is a homomorphism, if \(\psi(ab)=\psi(a)\psi(b) ,\forall a, b \in G\)
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Normal Subgroup means a subgroup which is closed under conjugate operation with any element in \(G\), i.e. \( \forall a \in G, aNa^{-1} \subset N\)
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Lagrange's Theorem
Definition of Equivalence relation "\(\sim\)" iff
- \(a\sim a\)
- \(a\sim b \Rightarrow b\sim a\)
- \(a\sim b, b\sim c \Rightarrow a\sim c\)
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Definition \([a]\), called class of \(a\), is defined by \([a]=\{ b\in S| b\sim a\}\)
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Lagrange's Theorem
\(G\) group and \(H\) the subgroup of \(G\), then \(|H|\) divides \(|G|\)
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