Definition:Index of a Subgroup

Definition

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

The index $\left[{G : H}\right]$ of $H$ (in $G$) is the number of left (or right) cosets of $G$ modulo $H$, or, the number of elements in the left (or right) coset space $G / H$, provided this number is finite (otherwise the index is infinite).

Also see

• Left and Right Coset Spaces are Equivalent, demonstrating that this definition is meaningful.

Also known as

Some sources use the notation $\left|{G : H}\right|$.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.3$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 25$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 39$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 43$
• John F. Humphreys: A Course in Group Theory (1996): $\S 5$: Definition $5.10$