Questions and Answers
Every permutation is a one to one function.
True
Every function is a permutation if and only if it is one to one.
False
Every function from a finite set onto itself must be one to one.
True
Every group G is isomorphic to a subgroup of S sub G.
Signup and view all the answers
Every subgroup of an abelian group is abelian.
Signup and view all the answers
Every element of a group generates a cyclic subgroup of the group.
Signup and view all the answers
The symmetric group S10 has 10 elements.
Signup and view all the answers
The symmetric group S3 is cyclic.
Signup and view all the answers
Sn is not cyclic for any n.
Signup and view all the answers
Every group is isomorphic to some group of permutations.
Signup and view all the answers
Every permutation is a cycle.
Signup and view all the answers
Every cycle is a permutation.
Signup and view all the answers
Every nontrivial subgroup H of S9 containing some odd permutation contains a transposition.
Signup and view all the answers
A5 has 120 elements.
Signup and view all the answers
Sn is not cyclic for any n greater than or equal to one.
Signup and view all the answers
A3 is a commutative group.
Signup and view all the answers
S7 is isomorphic to the subgroup of all those elements of S8 that leave the number 8 fixed.
Signup and view all the answers
S7 is isomorphic to the subgroup of all those elements of S8 that leave the number 5 fixed.
Signup and view all the answers
The odd permutations in S8 form a subgroup of S8.
Signup and view all the answers
Every subgroup of every group has left cosets.
Signup and view all the answers
The number of left cosets of a subgroup of a finite group divides the order of the group.
Signup and view all the answers
Every group of prime order is abelian.
Signup and view all the answers
One cannot have left cosets of a finite subgroup of an infinite group.
Signup and view all the answers
A subgroup of a group is a left cosets of itself.
Signup and view all the answers
Only subgroups of finite groups have left cosets.
Signup and view all the answers
An is of index 2 in Sn for n>1.
Signup and view all the answers
The theorem of Lagrange is a nice result.
Signup and view all the answers
Every finite group contains an element of every order that divides the order of the group.
Signup and view all the answers
Every finite cyclic group contains an element of every order that divides the order of the group.
Signup and view all the answers
If G1 and G2 are any groups, then G1 x G2 is always isomorphic to G2 x G1.
Signup and view all the answers
Computation in an external direct product of groups is easy if you know how to compute in each component group.
Signup and view all the answers
Groups of finite order must be used to form an external direct product.
Signup and view all the answers
A group of prime order could not be the internal direct product of two proper nontrivial subgroups.
Signup and view all the answers
Z2 x Z4 is isomorphic to Z8.
Signup and view all the answers
Z2 x Z4 is isomorphic to S8.
Signup and view all the answers
Z3 x Z8 is isomorphic to S4.
Signup and view all the answers
Every element in Z4 x Z8 has order 8.
Signup and view all the answers
The order of Z12 x Z15 is 60.
Signup and view all the answers
Zm x Zn has mn elements whether m and n are relatively prime or not.
Signup and view all the answers
Every abelian group of prime order is cyclic.
Signup and view all the answers
Every abelian group of prime power order is cyclic.
Signup and view all the answers
Z8 is generated by {4, 6}.
Signup and view all the answers
Z8 is generated by {4, 5, 6}.
Signup and view all the answers
Any two finitely generated abelian groups with the same Betti number are isomorphic.
Signup and view all the answers
Every abelian group of order divisible by 5 contains a cyclic subgroup of order 5.
Signup and view all the answers
Every abelian group of order divisible by 4 contains a cyclic subgroup of order 4.
Signup and view all the answers
Every abelian group of order divisible by 6 contains a cyclic subgroup of order 6.
Signup and view all the answers
Study Notes
Permutations and Functions
- Every permutation functions as a one-to-one function, maintaining unique mappings.
- Not every function qualifies as a permutation even if it is one-to-one, thus functions can exist without being permutations.
- A function that maps a finite set onto itself must be one-to-one to ensure every element is paired uniquely.
Groups and Subgroups
- Every group ( G ) can be represented as a subgroup within some symmetric group ( S_n ).
- All subgroups of abelian groups retain the property of being abelian.
- Any group element generates a cyclic subgroup, confirming the cyclical nature of group elements.
Symmetric Groups
- The symmetric group ( S_{10} ) does not contain 10 elements; it actually has ( 10! ) elements.
- The symmetric group ( S_3 ) is not cyclic, as it cannot be generated by a single element.
- For ( n \geq 1 ), ( S_n ) is not cyclic due to its structure.
Isomorphism in Groups
- Every group is isomorphic to some group of permutations, indicating a structural similarity.
- Not every permutation is a cycle; permutations can exhibit more complex structures.
- Every cycle qualifies as a permutation, reinforcing the relationship between cycles and permutations.
Cosets and Subgroups
- All subgroups possess left cosets, leading to a structured approach in group theory.
- The number of left cosets of a subgroup within a finite group divides the order of that group, establishing a foundational principle in group theory.
- Groups of prime order are inherently abelian.
Direct Products and Elements
- An isomorphic relationship exists between ( G_1 \times G_2 ) and ( G_2 \times G_1 ) for any groups ( G_1 ) and ( G_2 ).
- Working with external direct products is manageable if one understands computations within each component group.
- Finite-order groups are not a prerequisite for forming an external direct product, which can also involve infinite groups.
Group Orders and Structure
- A group of prime order cannot be expressed as the internal direct product of two proper nontrivial subgroups due to its simplicity.
- The group ( \mathbb{Z}_2 \times \mathbb{Z}_4 ) does not isomorphically match ( \mathbb{Z}_8 ) or ( S_8 ).
- The structure ( \mathbb{Z}_m \times \mathbb{Z}_n ) possesses ( mn ) elements regardless of whether ( m ) and ( n ) are relatively prime.
Cyclic Structures and Betti Numbers
- Every abelian group of prime order exhibits cyclic behavior, leading to simple structures.
- However, not all abelian groups of prime power order are cyclic, indicating complexity in higher powers.
- The group ( \mathbb{Z}_8 ) can be generated by the elements {4, 5, 6}, emphasizing its cyclic nature.
Subgroups Based on Order
- An abelian group whose order is divisible by 5 must contain at least one cyclic subgroup of order 5.
- Conversely, groups whose order is divisible by 4 or 6 do not necessarily contain cyclic subgroups of those respective orders, highlighting exceptions in subgroup structures.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge of Abstract Algebra with these true/false flashcards. Each card covers fundamental concepts such as permutations, functions, and groups. Challenge yourself and see how well you understand the intricate relationships in algebra!