Which property means “a*b is in G” for all a,b in G
A Identity
B Closure
C Inverse
D Commutative
Closure means performing the binary operation on any two elements of the set always gives an element still inside the same set, so the operation is well-defined on G.
A binary operation on a set must map
A S×S → S
B S → S×S
C S×S → ℤ
D ℤ → S
A binary operation takes an ordered pair of elements from the set and returns a single element of the same set. This ensures closure of the operation’s output set.
Associativity in a group requires
A ab = ba
B ae = a
C (ab)c = a(bc)
D aa⁻¹ = e
Associativity guarantees the way of placing brackets does not change the result of combining three elements. It is required for all elements in the group under the operation.
Identity element e satisfies
A ae = e
B ae = ea = a
C ea = e
D aea = e
The identity element leaves every group element unchanged when combined on either side. Existence of such an element is one of the defining axioms of a group.
Inverse of a in group means
A aa⁻¹ = a
B aa⁻¹ = e only
C a⁻¹a = a only
D aa⁻¹ = a⁻¹a = e
For each element a in a group, there must exist an element a⁻¹ such that combining them in either order gives the identity element, ensuring “undoing” of a.
An abelian group means
A (ab)c = a(bc)
B a⁻¹ exists
C ab = ba
D identity absent
Abelian (commutative) groups satisfy ab = ba for all elements. Associativity, identity, and inverses are already required for any group, not special to abelian groups.
In a group, identity element is
A Multiple always
B Unique
C Optional
D Same as inverse
A group cannot have two different identity elements. If e and f both act as identity, then e = ef = f, proving uniqueness by the group axioms.
Inverse of an element in a group is
A Two for each
B Not needed
C Same for all
D Unique
If b and c both behave as inverses of a, then b = be = b(ac) = (ba)c = ec = c. So each element has exactly one inverse.
A subgroup must be
A Any subset
B Only proper subset
C Nonempty subset
D Only infinite subset
A subgroup is a nonempty subset of a group that forms a group under the same operation. Nonempty is essential, because identity and inverses must exist inside it.
Subgroup test often checks
A closure and inverses
B commutativity only
C distributivity only
D ordering property
A common subgroup test: a nonempty subset H is a subgroup if for all a,b in H, the element ab⁻¹ is in H. This ensures closure and inverses together.
The trivial subgroup of G is
A G only
B {e}
C {0,1}
D empty set
The trivial subgroup contains only the identity element. It always exists in every group and satisfies all subgroup properties under the group operation.
A proper subgroup is
A H = G only
B H = empty set
C H ≠ G and H ⊂ G
D H has no identity
A proper subgroup is strictly smaller than the whole group but still a subgroup. It must contain the identity and inverses and be closed under the operation.
Order of a finite group means
A largest element value
B identity element
C number of subgroups
D number of elements
The order of a group is its cardinality. For finite groups, it is simply the total count of distinct elements in the group.
Order of an element a is
A largest n with aⁿ = e
B least n with aⁿ = e
C n where aⁿ = a
D n where a = e
The order of an element is the smallest positive integer n such that repeated operation returns the identity. If no such n exists, the element has infinite order.
A cyclic group is generated by
A one element
B two identities
C one element
D all elements together
A cyclic group has an element g such that every element of the group is gⁿ (or n·g in additive form). Such a g is called a generator.
In cyclic group ⟨g⟩, elements look like
A g+n
B gⁿ
C g/n
D n/g
In multiplicative notation, elements of the cyclic group generated by g are gⁿ for integers n. In additive notation, they are integer multiples n·g.
A group of prime order p is
A cyclic
B never cyclic
C always nonabelian
D has no subgroups
In a group of prime order p, any non-identity element must have order p by Lagrange’s theorem, so it generates the whole group, making the group cyclic.
Left coset of H by g is
A Hg
B H∪g
C gH
D H−g
A left coset is formed by multiplying every element of H on the left by g: gH = {gh : h in H}. This partitions the group into equal-sized pieces.
Right coset of H by g is
A gH
B Hg
C g+H
D H/g
A right coset is Hg = {hg : h in H}. In abelian groups, left and right cosets coincide, but in general groups they may differ.
Cosets of a subgroup in a finite group have
A random sizes
B size 1 always
C size equals index
D equal size
Each coset gH has the same number of elements as H because the map h → gh is a bijection from H to gH. Hence all cosets are equally sized.
Index [G:H] means
A number of cosets
B size of H
C number of cosets
D size of G only
The index of H in G is the number of distinct left cosets (equivalently right cosets) of H in G. For finite groups, [G:H] = |G|/|H|.
Lagrange’s theorem states that for finite G
A |G| divides |H|
B |H| divides |G|
C |H| = |G|
D |H| is prime
If H is a subgroup of a finite group G, then the order of H divides the order of G. This follows because G is partitioned into cosets of H.
A key consequence of Lagrange’s theorem is
A order of element divides |G|
B every group is cyclic
C every subgroup is normal
D all groups are abelian
The cyclic subgroup generated by an element a has size equal to the order of a. By Lagrange, that size divides |G|, so ord(a) divides |G|.
If |G| = 12, possible element order is
A 5
B 7
C 6
D 11
By Lagrange’s theorem, an element’s order must divide 12. Divisors include 1,2,3,4,6,12. So 6 is possible, while 5,7,11 are not.
If element a has order n, then aⁿ equals
A inverse of a
B identity
C a itself
D zero element
The order n of a is defined as the smallest positive integer with aⁿ = e. Thus aⁿ must be the identity element of the group.
A normal subgroup N satisfies
A gN ≠ Ng always
B N has prime order
C N contains no identity
D gN = Ng for all g
A subgroup N is normal if every left coset equals the corresponding right coset for all g in G. This condition makes quotient group construction well-defined.
Equivalent normality test is
A gN = N only
B Ng = N only
C gNg⁻¹ = N
D g²N = N
A subgroup N is normal in G if conjugating it by any g keeps it unchanged: gNg⁻¹ = N. This captures invariance under internal symmetry of G.
In an abelian group, every subgroup is
A cyclic
B normal
C of prime order
D trivial
If the group is abelian, then gH = Hg for all g and all subgroups H, because multiplication commutes. Hence every subgroup automatically satisfies normality.
Quotient group G/N uses elements as
A pairs (g,n)
B only generators
C elements of N only
D cosets of N
In the quotient group G/N, each element is a coset gN. The group operation is defined by (gN)(hN) = (gh)N, which is well-defined only if N is normal.
Coset multiplication in G/N is
A (gN)(hN) = (g+h)N
B (gN)(hN) = gh
C (gN)(hN) = (gh)N
D (gN)(hN) = N
The natural multiplication in the quotient uses representatives: multiply them in G and take the resulting coset. Normality ensures the result does not depend on chosen representatives.
The center Z(G) is the set of elements that
A generate whole group
B commute with all
C have prime order
D form only cosets
The center Z(G) contains all elements z such that zg = gz for every g in G. It is always a normal subgroup and measures how close G is to abelian.
A homomorphism φ: G→H must satisfy
A φ(ab)=φ(a)φ(b)
B φ(a+b)=φ(a)−φ(b)
C φ(ab)=φ(a)+φ(b)
D φ(a)=a always
A homomorphism preserves the group operation. This structure-preserving property allows transferring group behavior from G to H and is central to kernels, images, and quotient groups.
The kernel of homomorphism φ is
A {h : φ(h)=e}
B image of φ
C inverse set only
D {g : φ(g)=e}
The kernel is the set of elements in the domain mapped to the identity of the codomain. It is always a normal subgroup and describes how much information is “collapsed.”
The image of φ is
A kernel of φ
B subgroup of G only
C φ(G) subset of H
D set of cosets
The image is the set of all outputs φ(g) in H. It is always a subgroup of H. If the image equals all of H, the homomorphism is surjective.
A homomorphism is injective iff
A image is {e}
B kernel is {e}
C kernel equals G
D image equals G
If only the identity maps to identity, then distinct elements cannot map to the same output, so φ is one-to-one. This is a standard and very useful criterion.
First isomorphism theorem states
A kerφ ≅ imφ
B G ≅ kerφ
C H/kerφ ≅ G
D G/kerφ ≅ imφ
The theorem connects quotient groups and images: collapsing G by the kernel produces a group isomorphic to the image. It explains why kernels naturally lead to quotient structures.
Composition of homomorphisms is
A never a homomorphism
B only bijective
C a homomorphism
D only for rings
If φ and ψ preserve operation, then ψ∘φ also preserves operation. This closure under composition is important for building chains of structure-preserving maps in algebra.
An isomorphism is a homomorphism that is
A only surjective
B bijective
C only injective
D never injective
An isomorphism is a structure-preserving map that is both one-to-one and onto. It shows two algebraic structures are essentially the same up to renaming elements.
An endomorphism is a homomorphism
A from G to G
B from G to H only
C from H to G only
D from cosets to G
Endomorphism means a homomorphism from a structure to itself. It preserves operation within the same group and is used to study internal symmetries and transformations.
An automorphism is
A non-bijective map
B map to different group
C bijective endomorphism
D only kernel map
An automorphism is an isomorphism from a group to itself. It preserves structure and is reversible. Automorphisms form a group under composition, called Aut(G).
A ring requires under addition
A semigroup only
B no identity
C no inverses
D abelian group
In a ring, the set with addition must form an abelian group: associativity, identity (0), inverses, and commutativity. Multiplication is associative, and distributive laws connect both.
A ring with unity means
A has additive identity only
B has multiplicative identity
C multiplication commutative always
D no zero element
A ring with unity has an element 1 such that 1a = a1 = a for all a. This unity is different from the additive identity 0.
Zero divisors in a ring are nonzero a,b with
A a+b = 0
B a = b
C ab = 0
D a² = a
Zero divisors are nonzero elements whose product becomes zero. Their presence prevents cancellation in multiplication and is a key reason some rings are not integral domains.
An ideal I of ring R satisfies
A rI ⊆ I and Ir ⊆ I
B only addition closure
C only multiplication closure
D contains only units
An ideal is an additive subgroup that absorbs multiplication by ring elements. For all r in R and a in I, both ra and ar must lie in I (two-sided ideal).
Principal ideal in ℤ generated by n is
A ℤ/n
B nℤ
C ℤ×n
D {n} only
In integers, the ideal generated by n is all multiples of n: nℤ = {nk : k in ℤ}. Every ideal in ℤ is principal, which is a fundamental example.
Quotient ring R/I elements are
A only elements of I
B only units
C cosets a+I
D ordered pairs
The quotient ring uses additive cosets a+I as elements. Addition and multiplication are defined using representatives, and the ideal property ensures operations are well-defined.
An integral domain is a commutative ring with unity and
A no identity element
B multiplication nonassociative
C always finite
D no zero divisors
Integral domains have commutative multiplication, unity, and no zero divisors. This implies cancellation: if a≠0 and ab=ac, then b=c, making arithmetic behave like integers.
ℤ₆ is not an integral domain because
A it has no zero
B it has zero divisors
C it has no addition
D it is infinite
In ℤ₆, 2 and 3 are nonzero but 2·3 ≡ 0 (mod 6), so zero divisors exist. Therefore ℤ₆ fails the integral domain condition.
A field is a commutative ring with unity where
A zero has inverse
B no addition inverses
C every nonzero has inverse
D multiplication not closed
In a field, each nonzero element has a multiplicative inverse, making division possible (except by zero). This is stronger than integral domain and eliminates zero divisors automatically.
ℤₚ forms a field when p is
A prime
B composite
C even only
D multiple of 3
Modulo p arithmetic gives a field exactly when p is prime, because then every nonzero residue class has an inverse. For composite modulus, zero divisors appear and inverses fail.