Group theory

1. Group theory is a branch of abstract algebra that deals with the study of mathematical structures called groups. A group is a set equipped with an operation that combines any two elements to produce a third element in such a way that certain properties are satisfied, such as closure, associativity, identity element, and invertibility. Group theory has applications in various fields, including mathematics, physics, chemistry, and computer science. It provides a framework for understanding symmetry, transformations, and patterns, and has profound implications in the study of symmetry groups, group representations, and group actions.

What is the identity element of a group?

A) An even number in the group.
B) An element that is the largest in the group.
C) An element in the group such that when combined with any other element, the result is that other element.
D) An element that is the smallest in the group.

2. What does it mean for a group operation to be associative?

A) For all elements a, b, c in the group, (a * b) * c = a * (b * c).
B) For all elements a, b in the group, a * b = b * a.
C) For all elements a, b in the group, a = a * b.
D) For all elements a, b, c in the group, (a + b) * c = a * (b * c).

3. What is Lagrange's theorem in group theory?

A) In a finite group, the order of a subgroup divides the order of the group.
B) The largest element in a group.
C) The sum of all elements in a group equals zero.
D) A theorem about linear algebra.

4. What is an abelian group?

A) A group with only one element.
B) A group with no identity element.
C) A group where the operation is defined only for odd numbers.
D) A group where the group operation is commutative.

5. What does it mean for a group to be cyclic?

A) A group generated by a single element.
B) A group with no operation defined.
C) A group with no identity element.
D) A group where elements can have multiple inverses.

6. What is the definition of the center of a group?

A) The set of elements that commute with every element of the group.
B) The set of inverses of the group.
C) The largest element in the group.
D) The sum of all elements in a group.

7. What is the definition of the order of a group?

A) The smallest element in the group.
B) The number of elements in the group.
C) The largest element in the group.
D) The sum of all elements in the group.

8. What is the definition of an automorphism of a group?

A) A group of integers.
B) An isomorphism from a group to itself.
C) A group with no identity element.
D) A group with only one element.

9. What is the definition of a dihedral group?

A) A group with no identity element.
B) A group of integers.
C) The group of symmetries of a regular polygon.
D) A group with only one element.

10. What is a permutation group?

A) A group where the elements are permutations of a set and the group operation is composition of permutations.
B) A group with no identity element.
C) A group of integers.
D) A group with only one element.

11. What is the definition of a symmetric group?

A) A group of integers.
B) A group with only one element.
C) A group with no identity element.
D) The group of all permutations of a set.

12. What is the definition of the homomorphism between two groups?

A) A function between two groups that preserves the group structure.
B) The sum of all elements in a group.
C) The smallest element in the group.
D) The largest element in the group.

13. What is the definition of the quotient group?

A) The sum of all elements in a group.
B) A group with no identity element.
C) The group of cosets of a normal subgroup.
D) The largest element in the group.

14. What is the definition of an alternating group?

A) A group with no identity element.
B) The subgroup of the symmetric group consisting of even permutations.
C) A group of integers.
D) A group with only one element.

15. What does it mean for two groups to be isomorphic?

A) The sum of all elements in a group is the same.
B) The smallest element in the groups is the same.
C) The groups have the same structure, even if the elements may be labeled differently.
D) The largest element in the group is identical.

16. What is the Cayley's theorem in group theory?

A) Every group is isomorphic to a permutation group.
B) A theorem about linear algebra.
C) The sum of all elements in a group.
D) The largest element in a group.

17. What does the term 'conjugacy class' refer to in group theory?

A) A group with only one element.
B) A group of integers.
C) A group with no identity element.
D) A set of elements that are all conjugates of each other.

18. What is the definition of the commutator subgroup?

A) The subgroup generated by all commutators.
B) A group with no identity element.
C) The largest element in the group.
D) The sum of all elements in a group.

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