Abstract Algebra True/ False Questions

Every Integral domain contains a set of positive elements.

False - an Integral Domain does not necessarily come with an order, only an ordered integral domain would have a set of “positive” elements.

It is impossible to impose an order relation in the sec C of complex numbers.

True - There is no order on C that is compatible with field operations. C cannot be made into an ordered field.

In any ordered integral domain, the unity element e is a positive element.

True - in any ordered field or domain, 1 must be positive.

The set R of real numbers is an ordered integral domain.

True - R is a communative ring with unity, has no zero divisors, and has < which means it is an ordered integral domain.

The set of all integers is well ordered.

False - Z with the usual order is not well ordred cause it has no smallest element.

The field Q of rational numbers is an extension of the integral domain Z of integers.

True- Q contains Z as a subring, so Q is indeed an extension of Z

The field of quotients Q of an integral domain D contains D

False - D is not a subset of the set of equivalence classes.

The field R of real numbers is an extension of the integral domain Z of integers.

False - Z is not a field so R cannot be a field extension of Z

The field of quotients Q of an integral domain D contains a subring D' = {[x, eJ Ix E D, and e is the unity in D}.

True - elements of the form x/1 or [x,e] forms a subring isomorphic to D, which is in the field of quotients.

A field of quotients can be constructed from an arbitrary integral domain.

True- every integral domain has a field of fractions

An integral domain contains at least two elements.

True - an integral domain has a unity (1), so it must contain at least two elements 0 and 1

Every field is an integral domain

True - A field has no zero divisors, so it automatically satisfies the definition of an integral domain

Every integral domain is a field.

False - Take Z, not every element has a multiplicative inverse

If a set S is not an integral domain, then S is not a field.

True - Every field is an integral domain, so if something is not an integral domain it cannot be a field

Every ring is an abelian group with respect to the operations of addition and multiplication.

False - a ring is an abelian group under addition but not multiplication

Let R be a ring. The set { 0} is a subring of R with respect to the operations in R

True, {0} is the zero ring, which satisfies all ring axioms and is a valid subring

Let R be a ring. Then R is a subring of itself.

True - any structure is trivially a substructure of itself

Both E, the set of even integers, and Z - E, the set of odd integers, are subrings of the

set Z of all integers.

False - E is a subring, but the odd integers are not closed under addition

If one element in a ring R has a multiplicative inverse, then all elements in R must have multiplicative inverses

False - not all integers have inverses

Let x and y be elements in a ring R. If xy=0, then either x=0 or y=0.

False - only holds in integral domains

Let R be a ring with unity and S a subring (with unity) of R. Then R and S must have

the same unity elements.

False - a subring with unity can have a different unity

A unity exists in any commutative ring.

False - Plenty of rings have no identity

Any ring with unity must be commutative.

False - non-commutative rings with identity exist

Zn is a subring of Z, where n Ez+ and n>1.

False - Zn is a quotient ring, not a subring of Z.

The symmetries of any plane figure form a group under mapping composition.

True - Symmetries are closed under composition include the identity and each has an inverse - form a group

The regular pentagon possesses only rotational symmetry.

False - pentagon has 5 rotational symmetries and 5 reflections.

The regular hexagon possesses both rotational and reflective symmetry.

True - hexagon has 6 rotations and 6 reflections

The group Dn of symmetries for a regular polygon with n sides has order n.

False - has order 2n, n rotations and n reflections so order is 2n not n

The symmetric group S3 on 3 elements is the same as the group D3 of symmetries for an equilateral triangle. That is, S3 = D3.

True - both have 6 elements, structure is identical, thus it is isomorphic.

The symmetric group S4 on 4 elements is the same as the group D4 of symmetries for a square. That is, S4=D4

False - Not the same group, different orders and structure - no isomorphism

The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is, A4=D4.

False - different orders and structure, not the same group and not isomorphic

Every finite group G of order n is isomorphic to a subgroup of order n of the group

S(G) of all permutations on G.

True - S(G) is isomporhic to Sn. There exists a subgroup of S(G) of order n that is isomorphic to G

Every permutation can be written as a product of transpositions.

True - any permutation can be decomposed into 2-cycles

A permutation can be uniquely expressed as a product of transpositions.

False - expression is not unique, many different transposition products represent the same permutation

The product of cycles under mapping composition is a commutative operation.

False - cycles generally do not commute unless they are disjoint.

Disjoint cycles commute under mapping composition

True - disjoint cycles act on different elements, so composition does not matter.

The identity permutation can be expressed in more than one way

True - () or (1,2) (1,2), many different ways

Every permutation can be expressed as a product of disjoint cycles.

True - decomposition is unique up to order of cycles

An r-cycle is an even permutation if r is even and an odd permutation if r is odd.

False - r-cycles is the product of r-1 transpositions so if r-1 is odd then r is even

The set of all odd permutations in Sn is a subgroup of Sn

False - odd permutations are not closed under composition

The symmetric group Sn on n elements has order n

False - has order n!

A transposition leaves all elements except two fixed

True - A transposition swaps exactly 2 elements and fixes all others.

The order of an r-cycle is r.

True - applying an r cycle r times returns all elements to their original positions.

The mutually disjoint cycles of a permutation are the same as its orbits.

False - not the same objects.

Any two cyclic groups of the same order are isomorphic.

True - cyclic groups of the same order are all isomorphic

Any two abelian groups of the same order are isomorphic.

False - Z4 and Z2 x Z2 are both order 4 but not isomorphic

Any isomorphism is an automorphism.

False - only an automorphism if the domain and codomain are the same group

Any automorphism is an isomorphism.

True - any automorphism is an isomorphism

If two groups G and G ' have order 3, then G and G ' are isomorphic

True - Every group of prime order is cyclic, so any group of order 3 are cyclic and isomorphic

Any two groups of the same finite order are isomorphic

False - groups of order 4

Two groups can be isomorphic even though their group operations are different.

True - isomorphism requires a bijective map preserving the operation, the actual operation symbols may differ

The relation of being isomorphic is an equivalence relation on a collection of groups

True - it is reflexive, symmetric and transitive.

The order of the identity element in any group is 1.

True - no smaller positive power works

Every cyclic group is abelian.

True - commutativity is satisfied.

Every abelian group is cyclic.

False - can be abelian without being cyclic

If a subgroup H of a group G is cyclic, then G must be cyclic.

False - a cyclic subgroup does not mean the entire thing is cyclic

Whether a group G is cyclic or not, each element a of G generates a cyclic sub­ group.

True - always cyclic by definition

Every subgroup of a cyclic group is cyclic.

True - standard result by group theory

If there exists an m E Z+such that am = e, where ais an element of a group G, then lal = m.

False - lal has to be the smallest positive integer

Any group of order 3 must be cyclic.

True - groups of prime order are always cyclic

Any group of order 4 must be cyclic.

False - Z2 x Z2 has order 4 but not cyclic

Let a be an element of a group G. Then (a) = (a-1).

True - powers of inverse a produce the same set of elements as powers of a just in reverse order

Every group G contains at least two subgroups.

True - every group has at least the trivial group {e} and the group itself

The identity element in a subgroup Hof a group G must be the same as the identity element in G.

True - the identity of a subgroup is always the identity of the parent group

An element x in H has an inverse x-1 in H that may be different than its inverse in G.

False - the inverse of x in H must be the same as in G because the identity is the same and inverses are unique

The generator of a cyclic group is unique

False - a cyclic group of order n has phi n generators

Any subgroup of an abelian group is abelian.

True - commutativity is inherited by subgroups

If a subgroup H of a group G is abelian, then G must be abelian.

False - the subgroup being abelian does not mean the whole group is abelian

The relation R on the set of all groups defined by HRK if and only if H is a subgroup ofK is an equivalence relation.

False - not symmetric

The empty set 0 is a subgroup of any group G

False - subgroup must contain the identity and empty set does not

Suppose G is a group with respect to ® and H� G is a group with respect to a different

binary operation@. Then His a subgroup of G.

False - using a different operation does not make it a subgroup

Any group of order 3 has no nontrivial subgroups.

True - groups of prime order are cyclic and only have the trivial subgroup {e} and the whole group

Z5 under addition modulo 5 is a subgroup of the group Z under addition.

False - not closed under addition, not a subgroup

A group may have more than one identity element.

False - identity element is unique

An element in a group may have more than one inverse

False - every element in a group has a unique inverse

Let x, y, and z be elements of a group G. Then (xyz)-1=x-•y-•z-•.

False - order is reversed for inverses

In a Cayley table for a group, each element appears exactly once in each row.

True - each row is a permutation of the group elements

The Generalized Associative Law applies to any group, no matter what the group operation is.

True - associativity is part of the group axioms

If x2= e for at least one x in a group G, then x2=e for all xE G.

False - having order 2 does not mean every element will have order 2

The set Z of all integers is a nonabelian group with respect to subtraction.

False - not associative

The set R - { 0} of nonzero real numbers is a nonabelian group with respect to division.

False - not associative

The identity element in a group G is its own inverse.

True e x e =e so e^-1=e

If G is an abelian group, then x-1 = x for all x inG.

False - inverses exist but not necessarily equal to the element

Let G be a group that is not abelian. Then xy =F yx for all x and y in G.

False - non abelian means some pairs do not commute but not necessarily all.

The set of all nonzero elements in Zs is an abelian group with respect to multiplication.

False - usually only if s is prime

The Cayley table for a group will always be symmetric with respect to the diagonal from upper left to lower right.

False - symmetry only occurs if the group is abelian

If a set is closed with respect to the operation, then every element must have an inverse.

False - closure alone does not guarantee inverses.

The nonzero elements of Mmxn (R) form a group with respect to matrix multiplication.

False - not all nonzero matrices are invertible

The nonzero elements of Mn (R) form a group with respect to matrix multiplication

False - fails closure

The invertible elements of Mn (R) form an abelian group with respect to matrix multi­ plication.

False - matrix multiplication is not commutative for n is greater than 1 so the group is non abelian

2EZ4.

False - not an equivalence class

[7] EZ4

True - by modulo 4

Every element [a] in Zn has an additive inverse.

True

Every element [ a] -=F [O] in Zn has a multiplicative inverse.

False - only [a] coprime to n has a multiplicative inverse

[a][b]= [O] implies either [a]= [O] or [b]= [O] in Zn.

False, zero divisors exist if n is not prime

[a][x]= [a][y]and [a]-=F [O] implies [x]= [y] in Zn.

False - cancellation may fail if it is not invertible

[a]= [l] implies (a,n)= 1 in Zn

True - implies gcd =1

(a, n) = 1 implies [a]= [l] in Zn

False - being co prime does not mean a is congruent to 1 mod n

1a=b(modn)implies ac=be (mod nc) for cEZ+.

True

a=b (modn)and cln imply a= b (mod c)for c E z+

True