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Assessment
What is the definition of a group in abstract algebra?
A set with a single binary operation that satisfies closure, associativity, has an identity element, and every element has an inverse.
A collection of vectors in a vector space that can be scaled and added together.
A set of numbers under addition or multiplication where all operations satisfy the commutative property.
A system of equations that has at least one solution.
Which of the following statements about normal subgroups is true?
Every subgroup of a finite group is normal.
A subgroup H of a group G is normal if and only if gHg^(-1) = H for all g in G.
Normal subgroups can only exist in abelian groups.
Normal subgroups must always be of index two in the group.
What is the order of an element in a group?
The number of elements in the group.
The smallest positive integer n such that g^n = e, where e is the identity element.
The total number of cosets of a subgroup in a group.
The number of distinct elements in a subgroup generated by that element.
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