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.
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Every subgroup of an abelian group is abelian.
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Every element of a group generates a cyclic subgroup of the group.
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The symmetric group S10 has 10 elements.
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The symmetric group S3 is cyclic.
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Sn is not cyclic for any n.
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Every group is isomorphic to some group of permutations.
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Every permutation is a cycle.
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Every cycle is a permutation.
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Every nontrivial subgroup H of S9 containing some odd permutation contains a transposition.
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A5 has 120 elements.
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Sn is not cyclic for any n greater than or equal to one.
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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.
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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.
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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.
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Every group of prime order is abelian.
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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.
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Only subgroups of finite groups have left cosets.
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An is of index 2 in Sn for n>1.
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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.
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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.
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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.
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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.
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Z2 x Z4 is isomorphic to S8.
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Z3 x Z8 is isomorphic to S4.
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Every element in Z4 x Z8 has order 8.
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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.
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Every abelian group of prime order is cyclic.
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Every abelian group of prime power order is cyclic.
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Z8 is generated by {4, 6}.
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Z8 is generated by {4, 5, 6}.
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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.
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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.
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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!
