Abelian group

In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that

Multiplication table

To verify that a certain finite group is indeed abelian, a table (matrix) can be drawn up in the similar fashion to a multiplication table, where, if the group is G = {g1 = e, g2, ..., gn} under the operation ⋅, the (i, j)'th entry of this table contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal (i.e. if the matrix is a symmetric matrix).

Related Topics:
Finite group - Multiplication table - If and only if - Symmetric matrix

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This is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the (i, j)'th entry of the table equals the (j, i)'th entry - i.e. the table is symmetric about the main diagonal.

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