Cayley Table

Definition

A Cayley table is a mathematical table used to represent the operation of a finite group in abstract algebra. It provides a visual and structured way to show how elements of the group combine under the group's binary operation, making it easier to verify group properties such as closure, associativity, and the existence of identity and inverse elements.

5 Must Know Facts For Your Next Test

1. The Cayley table is square-shaped, with both its rows and columns representing the elements of the group.
2. Each entry in the table corresponds to the result of combining the row and column elements using the group's binary operation.
3. The diagonal of a Cayley table shows the identity element when each element is combined with itself, demonstrating self-identity.
4. Cayley tables can be used to identify whether a given set with a specific operation forms a group by checking for group properties visually.
5. If two groups are isomorphic, their Cayley tables will have the same structure but possibly different entries.

Review Questions

• How does a Cayley table help verify the properties of a group?
  • A Cayley table visually demonstrates how each element in a group interacts with others through its binary operation. By checking each entry for closure, associativity can be inferred if the table is consistent. Additionally, identifying an entry that serves as an identity element and ensuring every element has an inverse in the table are crucial steps in verifying all group properties.
• What role does the Cayley table play in understanding group isomorphism?
  • The Cayley table illustrates the structure of a group's operation, enabling one to compare different groups for isomorphism. Two groups are isomorphic if there exists a bijective mapping between their elements that preserves the binary operation. By analyzing their Cayley tables, one can identify if they exhibit similar structural patterns, which indicates potential isomorphism.
• Evaluate how Cayley tables can be constructed for non-abelian groups and what implications this has on group structure.
  • Constructing a Cayley table for non-abelian groups shows that the entries may not be symmetric across the diagonal due to the lack of commutativity in operations. This indicates that changing the order of operations leads to different results. By analyzing these tables, one can gain insights into how non-abelian groups behave differently from abelian ones and understand more about their internal structure and complexities.

Related terms

Group:

A set combined with an operation that satisfies four fundamental properties: closure, associativity, identity element, and invertibility.

Binary Operation: A calculation that combines two elements from a set to produce another element from the same set.

Identity Element: An element in a group that, when combined with any other element in the group, results in that same element.