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Questions and Answers
Every permutation is a one to one function.
Every permutation is a one to one function.
True
Every function is a permutation if and only if it is one to one.
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.
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.
Every group G is isomorphic to a subgroup of S sub G.
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Every subgroup of an abelian group is abelian.
Every subgroup of an abelian group is abelian.
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Every element of a group generates a cyclic subgroup of the group.
Every element of a group generates a cyclic subgroup of the group.
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The symmetric group S10 has 10 elements.
The symmetric group S10 has 10 elements.
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The symmetric group S3 is cyclic.
The symmetric group S3 is cyclic.
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Sn is not cyclic for any n.
Sn is not cyclic for any n.
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Every group is isomorphic to some group of permutations.
Every group is isomorphic to some group of permutations.
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Every permutation is a cycle.
Every permutation is a cycle.
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Every cycle is a permutation.
Every cycle is a permutation.
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Every nontrivial subgroup H of S9 containing some odd permutation contains a transposition.
Every nontrivial subgroup H of S9 containing some odd permutation contains a transposition.
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A5 has 120 elements.
A5 has 120 elements.
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Sn is not cyclic for any n greater than or equal to one.
Sn is not cyclic for any n greater than or equal to one.
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A3 is a commutative group.
A3 is a commutative group.
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S7 is isomorphic to the subgroup of all those elements of S8 that leave the number 8 fixed.
S7 is isomorphic to the subgroup of all those elements of S8 that leave the number 8 fixed.
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S7 is isomorphic to the subgroup of all those elements of S8 that leave the number 5 fixed.
S7 is isomorphic to the subgroup of all those elements of S8 that leave the number 5 fixed.
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The odd permutations in S8 form a subgroup of S8.
The odd permutations in S8 form a subgroup of S8.
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Every subgroup of every group has left cosets.
Every subgroup of every group has left cosets.
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The number of left cosets of a subgroup of a finite group divides the order of the group.
The number of left cosets of a subgroup of a finite group divides the order of the group.
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Every group of prime order is abelian.
Every group of prime order is abelian.
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One cannot have left cosets of a finite subgroup of an infinite group.
One cannot have left cosets of a finite subgroup of an infinite group.
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A subgroup of a group is a left cosets of itself.
A subgroup of a group is a left cosets of itself.
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Only subgroups of finite groups have left cosets.
Only subgroups of finite groups have left cosets.
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An is of index 2 in Sn for n>1.
An is of index 2 in Sn for n>1.
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The theorem of Lagrange is a nice result.
The theorem of Lagrange is a nice result.
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Every finite group contains an element of every order that divides the order of the group.
Every finite group contains an element of every order that divides the order of the group.
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Every finite cyclic group contains an element of every order that divides the order of the group.
Every finite cyclic group contains an element of every order that divides the order of the group.
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If G1 and G2 are any groups, then G1 x G2 is always isomorphic to G2 x G1.
If G1 and G2 are any groups, then G1 x G2 is always isomorphic to G2 x G1.
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Computation in an external direct product of groups is easy if you know how to compute in each component group.
Computation in an external direct product of groups is easy if you know how to compute in each component group.
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Groups of finite order must be used to form an external direct product.
Groups of finite order must be used to form an external direct product.
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A group of prime order could not be the internal direct product of two proper nontrivial subgroups.
A group of prime order could not be the internal direct product of two proper nontrivial subgroups.
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Z2 x Z4 is isomorphic to Z8.
Z2 x Z4 is isomorphic to Z8.
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Z2 x Z4 is isomorphic to S8.
Z2 x Z4 is isomorphic to S8.
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Z3 x Z8 is isomorphic to S4.
Z3 x Z8 is isomorphic to S4.
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Every element in Z4 x Z8 has order 8.
Every element in Z4 x Z8 has order 8.
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The order of Z12 x Z15 is 60.
The order of Z12 x Z15 is 60.
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Zm x Zn has mn elements whether m and n are relatively prime or not.
Zm x Zn has mn elements whether m and n are relatively prime or not.
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Every abelian group of prime order is cyclic.
Every abelian group of prime order is cyclic.
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Every abelian group of prime power order is cyclic.
Every abelian group of prime power order is cyclic.
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Z8 is generated by {4, 6}.
Z8 is generated by {4, 6}.
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Z8 is generated by {4, 5, 6}.
Z8 is generated by {4, 5, 6}.
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Any two finitely generated abelian groups with the same Betti number are isomorphic.
Any two finitely generated abelian groups with the same Betti number are isomorphic.
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Every abelian group of order divisible by 5 contains a cyclic subgroup of order 5.
Every abelian group of order divisible by 5 contains a cyclic subgroup of order 5.
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Every abelian group of order divisible by 4 contains a cyclic subgroup of order 4.
Every abelian group of order divisible by 4 contains a cyclic subgroup of order 4.
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Every abelian group of order divisible by 6 contains a cyclic subgroup of order 6.
Every abelian group of order divisible by 6 contains a cyclic subgroup of order 6.
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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.
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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!
