Abelian Groups
UG
BMLabs Mathematics
REPOSITORY
SCAN TO READ FULL UNIT
Abstract AlgebraIntroduction to GroupsAbelian Groups
Groups of Exponent Two
After proving the inverse product criterion, we now study a common sufficient condition for a group to be abelian. Suppose every element satisfies . Then every element is its own inverse. This strong condition forces commutativity. Groups with this property appear naturally in examples such as the Klein four group and vector spaces over the field with two elements. In this lesson, students will prove that a group in which every element has square equal to the identity must be abelian.
DEFINITION : Exponent Two Condition
Let be a group with identity element . We say that satisfies the if
Equivalently, every element of is its own inverse.
BMLABS MATHEMATICS REPOSITORY
mathematics.bmlabs.co.in
Author
Dr. Bivash Majumder
Assistant Professor in Mathematics
Prabhat Kumar College, Contai
Groups of Exponent Two
After proving the inverse product criterion, we now study a common sufficient condition for a group to be abelian. Suppose every element satisfies . Then every element is its own inverse. This strong condition forces commutativity. Groups with this property appear naturally in examples such as the Klein four group and vector spaces over the field with two elements. In this lesson, students will prove that a group in which every element has square equal to the identity must be abelian.