Show that (\mathbb{R} - \{0\}, *) is an abelian (commutative) group where * is defined as a * b =...
Question:
Show that {eq}(\mathbb{R} - \{0\}, *) {/eq} is an abelian (commutative) group where {eq}* {/eq} is defined as {eq}a * b = \dfrac{ab}{3} {/eq}.
Abelian Group:
A group is a number of elements that satisfy a binary operation {eq}* {/eq} and holds the properties of closure, associativity, identity, and inverse. A group where for two elements {eq}a {/eq} and {eq}b {/eq}, the commutative property {eq}a * b = b * a {/eq} is satisfied, is called an Abelian group.
Answer and Explanation: 1
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- The group is {eq}\left( \mathbb{R}-\{0\},* \right) {/eq}.
- The composition is {eq}a * b = \dfrac{{ab}}{3} {/eq}.
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Chapter 19 / Lesson 6Abstract algebra involves groups, but also subgroups which can be tested to determine if they belong. Learn about different types of subgroups through proper and trivial subgroups, as well as the meaning of centers of groups in abstract algebra.