2.4 Cayley's Theorem and its Applications

Cayley's Theorem is a game-changer in group theory. It shows that every group can be seen as a bunch of permutations, making abstract groups more concrete. This lets us study groups by looking at how they shuffle things around.

This theorem fits perfectly into our study of group homomorphisms and isomorphisms. It gives us a powerful way to represent any group as permutations, helping us understand group structure and properties better.

Cayley's Theorem

Statement and Proof

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• Cayley's Theorem states every group G is isomorphic to a subgroup of the symmetric group on G
• Proof constructs a bijective homomorphism from G to a subgroup of Sym(G)
• Key step defines the left regular representation mapping each element g of G to permutation λg of G
• Left regular representation given by λg(x) = gx for all x in G
• Mapping g → λg shown to be an injective homomorphism from G to Sym(G)
• Group action concept crucial for understanding mechanism behind Cayley's Theorem
• Proof concludes by demonstrating image of G under homomorphism forms subgroup of Sym(G) isomorphic to G
• Isomorphism preserves group structure including operation and identity element
• Example: For cyclic group $C_3 = \{e, a, a^2\}$, left regular representation maps:
  • e → (1)
  • a → (123)
  • $a^2$ → (132)

Implications and Significance

• Provides powerful tool for representing abstract groups as concrete permutation groups
• Every finite group of order n isomorphic to subgroup of Sn (symmetric group on n elements)
• Allows analysis of group structure through study of permutations and cycle decompositions
• Used to prove every group of prime order cyclic by examining possible subgroups of corresponding symmetric group
• Facilitates classification of small groups by embedding them in symmetric groups of appropriate sizes
• Applies to constructing counterexamples in group theory disproving false conjectures about group properties
• Example: Group of order 4 embedded in S4 as subgroup {(1), (12)(34), (13)(24), (14)(23)}

Analyzing Groups with Cayley's Theorem

Structure Analysis

• Representation allows examination of group elements as permutations
• Cycle decompositions of permutations reveal order and structure of group elements
• Subgroup structure of original group reflected in subgroup structure of permutation representation
• Conjugacy classes in original group correspond to conjugacy classes in permutation representation
• Center of group identifiable through elements with specific permutation properties
• Example: In S4, subgroup {(1), (12)(34), (13)(24), (14)(23)} isomorphic to Klein four-group

Properties and Classification

• Theorem implies every group of order n divides n! as it must be isomorphic to subgroup of Sn
• Used to determine possible orders of elements in group of given size based on cycle structures in corresponding symmetric group
• Aids in proving groups of certain orders have specific properties
• Example: Proves every group of order $p^2$ (p prime) abelian by analyzing its representation in $S_{p^2}$
• Facilitates construction of Cayley tables for abstract groups by representing them as permutation groups
• Useful in computational group theory for representing and manipulating groups as permutation groups

Groups and Permutation Representations

Regular Permutation Representation

• Homomorphism from G to Sym(G) defined by left multiplication action of G on itself
• Preserves group structure mapping group elements to permutations reflecting group's operation
• Kernel always trivial ensuring representation faithful (injective)
• Image of G under representation forms subgroup of Sym(G) isomorphic to G
• Allows study of abstract group properties through concrete permutation cycles
• Example: For group $\{e, a, b, ab\}$, regular permutation representation:
  • e → (1)
  • a → (12)(34)
  • b → (13)(24)
  • ab → (14)(23)

Visualizing Group Structure

• Permutation cycles illustrate how group elements act on each other
• Cycle lengths correspond to orders of group elements
• Fixed points in permutations indicate subgroup structure
• Composition of permutations mirrors group multiplication
• Inverse elements represented by inverse permutations
• Example: In S3, (123) and (132) are inverse permutations corresponding to inverse elements in original group

Applying Cayley's Theorem to Problems

Problem-Solving Strategies

• Embed given group into appropriate symmetric group
• Analyze cycle structures to determine element orders and subgroup properties
• Use permutation representation to construct counterexamples or prove theorems
• Apply theorem to show certain group properties must hold based on order
• Utilize permutation representation to visualize abstract group concepts
• Example: Prove group of order 5 cyclic by showing it must be isomorphic to subgroup of S5 generated by 5-cycle

Advanced Applications

• Construct examples of groups with specific properties by identifying appropriate subgroups of symmetric groups
• Use in computational group theory to efficiently represent and manipulate abstract groups
• Apply to study automorphism groups by examining permutations preserving group structure
• Analyze normal subgroups through permutations commuting with all elements of representation
• Investigate group actions on sets by composing regular representation with other group actions
• Example: Construct non-abelian group of order 8 as subgroup of S8 generated by (1234)(5678) and (15)(26)(37)(48)