Explanation: In any group, if you multiply both sides by a⁻¹ on the left, ab=ac becomes b=c. This “left cancellation” follows from existence of inverses and associativity.
A semigroup is a set with
A identity element
B inverses for all
C associative operation
D commutative law
Explanation: A semigroup requires only a closed associative binary operation. Identity and inverses are not required. So it is weaker than a monoid and much weaker than a group.
A monoid is a semigroup with
A inverse element
B identity element
C commutative law
D zero divisors
Explanation: A monoid is a semigroup that additionally has an identity element for the operation. Inverses are not required, so not every monoid is a group.
In additive group (ℤ,+), identity is
A 1
B −1
C ∞
D 0
Explanation: For integers under addition, adding 0 keeps any integer unchanged: a+0=0+a=a. Hence 0 is the additive identity, and each a has inverse −a.
In multiplicative group (ℝ{0},×), identity is
A 0
B −1
C 1
D 2
Explanation: For nonzero real numbers under multiplication, 1 leaves every element unchanged: a·1=1·a=a. Each nonzero a has inverse 1/a, so it forms a group.
Inverse of a in (ℤ,+) is
A −a
B 1/a
C a²
D |a|
Explanation: In an additive group, inverse means a+(inverse)=0. For integers, a+(−a)=0. The multiplicative inverse 1/a is not generally an integer, so it is not used here.
Inverse of a in (ℝ{0},×) is
A −a
B a+1
C 1/a
D a−1
Explanation: In a multiplicative group, inverse satisfies a·a⁻¹=1. For nonzero real a, a·(1/a)=1. This is why 0 is excluded from the multiplicative group of reals.
If group operation is “+”, order of element a means
A least n with aⁿ=1
B least n with a=0
C number of subgroups
D least n with n·a=0
Explanation: In additive notation, repeating the operation gives n·a = a+a+…+a. The order is the smallest positive n making n·a equal the additive identity 0.
In ℤₙ under addition, every element has
A a multiplicative inverse
B zero divisors only
C an additive inverse
D no identity
Explanation: (ℤₙ,+) is always an abelian group. For any class [a], the inverse is [n−a], since [a]+[n−a]=[0]. Multiplicative inverses exist only for units.
In ℤₙ, element [a] has multiplicative inverse iff
A gcd(a,n)=1
B a is even
C a is prime
D n is even
Explanation: [a] is a unit mod n exactly when a and n are coprime. Then there exists x with ax≡1 (mod n). If gcd(a,n)≠1, no such inverse exists.
A cyclic group of order n has exactly
A n generators always
B two identities
C n elements
D no subgroups
Explanation: “Order n” means the group contains n elements. In a cyclic group, the number of generators is φ(n), not always n, and it has many subgroups for divisors of n.
Number of generators of cyclic group of order n equals
A n−1 always
B φ(n)
C n always
D 2 always
Explanation: In a cyclic group ⟨g⟩ of order n, g^k is a generator precisely when gcd(k,n)=1. The count of such k is Euler’s totient function φ(n).
In a cyclic group of order 8, number of generators is
A 4
B 8
C 2
D 6
Explanation: For n=8, φ(8)=8(1−1/2)=4. The generators correspond to exponents k in {1,3,5,7} that are coprime to 8.
Subgroup generated by element a is
A [a]
B ker(a)
C ⟨a⟩
D im(a)
Explanation: ⟨a⟩ means the smallest subgroup containing a. It consists of all powers aⁿ (or multiples n·a). Its size equals the order of a.
If ord(a)=m, then |⟨a⟩| equals
A m
B |G|
C m²
D 2m
Explanation: The subgroup generated by a contains exactly the distinct powers of a until identity appears. By definition, ord(a)=m is the first time a^m=e, so there are m elements.
In finite group, size of each left coset gH is
A |G|
B [G:H]
C |H|
D |G|−|H|
Explanation: Mapping h → gh is a bijection from H to gH. So every coset has exactly as many elements as H, which is the key idea behind Lagrange’s theorem.
If |G|=20 and |H|=5, index [G:H] is
A 5
B 10
C 15
D 4
Explanation: For finite groups, [G:H]=|G|/|H|. Here 20/5=4. This means there are exactly 4 distinct cosets of H in G.
If [G:H]=2, then H is
A trivial
B cyclic only
C normal
D prime order
Explanation: Any subgroup of index 2 is normal because there are only two cosets: H and its complement coset. Left and right cosets must match since both partitions have two blocks.
A typical application of Lagrange is to show
A no element order 5 in |G|=12
B every group abelian
C all rings fields
D all subgroups cyclic
Explanation: If an element had order 5, then 5 must divide |G| by Lagrange. Since 5 does not divide 12, such an element cannot exist in any group of order 12.
Fermat’s little theorem is about
A a^p≡0 mod p
B a^(p−1)≡1 mod p
C a+b≡ab mod p
D p divides a always
Explanation: For prime p and a not divisible by p, Fermat says a^(p−1)≡1 (mod p). It comes from the fact that nonzero residues mod p form a group of size p−1.
Euler’s theorem generalizes Fermat as
A a^n≡1 mod n
B a^(n−1)≡1 mod n
C a^φ(n)≡1 mod n
D a^φ(n)≡0 mod n
Explanation: If gcd(a,n)=1, then a belongs to the unit group (ℤₙ)× with size φ(n). Group theory implies a^φ(n) equals identity, giving Euler’s theorem in modular arithmetic.
A subgroup N is normal if cosets satisfy
A gN=Ng for all g
B gN≠Ng for all g
C N has no inverses
D N is cyclic
Explanation: Normality means left and right cosets coincide for every g. This property is exactly what makes multiplication of cosets well-defined in the quotient group G/N.
Conjugate of element x by g is
A g⁻¹xg⁻¹
B xgg⁻¹
C g+x−g
D gxg⁻¹
Explanation: Conjugation transforms x to gxg⁻¹. Studying conjugates helps detect normal subgroups, centers, and class structure in groups, especially in non-abelian cases.
Kernel of group homomorphism is always
A cyclic subgroup
B trivial subgroup only
C normal subgroup
D not a subgroup
Explanation: The kernel is closed under operation and inverses, so it is a subgroup. It is also invariant under conjugation, which makes it normal, allowing quotient group G/kerφ.
If φ is onto, then image of φ equals
A H
B G
C kerφ
D {e}
Explanation: Surjective means every element of codomain H is hit by some g in G, so imφ=H. The kernel describes elements mapping to identity, not surjectivity.
If φ is one-to-one, then kernel is
A H
B G
C {e}
D nontrivial always
Explanation: Injective homomorphisms map distinct elements to distinct outputs. If any non-identity mapped to identity, it would collide with identity’s image, so kernel must be only the identity.
Second isomorphism theorem relates
A G/kerφ ≅ imφ
B HK/K ≅ H/(H∩K)
C G/N ≅ N
D H ≅ kerφ
Explanation: For subgroup H and normal K, the theorem compares the “combined” subgroup HK modulo K with H modulo intersection H∩K. It is useful for simplifying quotient structures.
Third isomorphism theorem states
A G ≅ G/N
B K ≅ N
C (G/N)/(K/N) ≅ G/K
D G/K ≅ K/G
Explanation: If N ⊆ K are normal in G, then factoring by N and then by K/N is the same as factoring directly by K. It helps organize chains of normal subgroups.
In a ring, distributive law means
A (ab)c=a(bc) only
B ab=ba always
C a+a=a always
D a(b+c)=ab+ac
Explanation: Rings connect addition and multiplication using distributive laws. Both left and right distributivity hold, ensuring multiplication “spreads” over addition, like ordinary arithmetic.
In any ring, additive identity is called
A 0
B 1
C e
D u
Explanation: The element 0 satisfies a+0=0+a=a for all a. It is required because ring addition forms an abelian group, so identity and inverses must exist.
In any ring, additive inverse of a is
A 1/a
B a²
C −a
D a+1
Explanation: Since (R,+) is an abelian group, every element a has an additive inverse −a with a+(−a)=0. This is independent of whether multiplicative inverses exist.
A commutative ring means
A ab=ba for all
B a+b≠b+a always
C has no unity
D has zero divisors
Explanation: Commutative ring refers to commutativity of multiplication. Addition is already commutative by ring definition. Commutativity does not guarantee lack of zero divisors.
In ℤ, the ideal generated by 1 is
A 0
B 2ℤ
C ℤ
D ℤ₂
Explanation: (1) in ℤ equals 1·ℤ, which is all integers. This shows 1 generates the whole ring as an ideal. It matches the idea that gcd(1,n)=1 gives full ideal.
In ℤ, intersection of ideals mℤ and nℤ equals
A gcd(m,n)ℤ
B lcm(m,n)ℤ
C (m+n)ℤ
D mn+ℤ
Explanation: A number is in both mℤ and nℤ iff it is a multiple of both m and n, i.e., a multiple of lcm(m,n). So intersection corresponds to least common multiple.
In ℤ, sum of ideals mℤ + nℤ equals
A lcm(m,n)ℤ
B mnℤ
C gcd(m,n)ℤ
D (m−n)ℤ
Explanation: The set {mx+ny} equals dℤ where d=gcd(m,n). This is Bézout’s identity in ideal form. So adding ideals corresponds to greatest common divisor in integers.
A maximal ideal I has property that R/I is
A a ring with zero divisors
B always trivial
C not a ring
D a field
Explanation: An ideal is maximal if there is no larger proper ideal. A key characterization: R/I is a field exactly when I is maximal. This is a standard bridge between ideals and fields.
A prime ideal P has property that R/P is
A an integral domain
B always a field
C never commutative
D not a ring
Explanation: In a commutative ring with unity, P is prime iff whenever ab in P then a in P or b in P. Equivalently, the quotient ring R/P has no zero divisors.
In ℤ, prime ideals are
A nℤ for any n
B {0,1}
C pℤ for prime p
D only 0 ideal
Explanation: In ℤ, ideals are nℤ. The ideal nℤ is prime exactly when n is prime (up to sign). Then ℤ/nℤ becomes an integral domain precisely for prime n.
In ℤ, maximal ideals are
A nℤ for any n
B only 0 ideal
C only ℤ itself
D pℤ for prime p
Explanation: ℤ/pℤ is a field exactly when p is prime, so pℤ is maximal. If n is composite, ℤ/nℤ has zero divisors, so nℤ is not maximal.
Characteristic of a ring is the smallest n with
A n·0 = 1
B n·a = a for all
C n·1 = 0
D n² = 0
Explanation: The characteristic is the least positive n such that adding 1 to itself n times gives 0. If no such n exists, the characteristic is 0, like in ℤ, ℚ, ℝ.
Characteristic of ℤ is
A 0
B 1
C 2
D prime only
Explanation: In ℤ, adding 1 repeatedly never gives 0, so no positive n satisfies n·1=0. Therefore the characteristic is 0. This matches usual integer arithmetic without wrap-around.
Characteristic of ℤₙ is
A 0
B 1
C φ(n)
D n
Explanation: In ℤₙ, n·[1]=[n]=[0]. No smaller positive number does this, so the characteristic is n. This reflects modular wrap-around after n steps.
Cancellation law in integral domain means
A ab=ba always
B a≠0, ab=ac ⇒ b=c
C a+a=a always
D 0 has inverse
Explanation: In an integral domain there are no zero divisors, so if a(b−c)=0 and a≠0, then b−c=0. This provides cancellation like integers.
A field has zero divisors
A always
B sometimes
C never
D only finite
Explanation: In a field, every nonzero element has an inverse. If ab=0 with a≠0, multiply by a⁻¹ to get b=0. So fields cannot have nonzero zero divisors.
A finite field must have size
A p^n
B n!
C 2n
D prime only
Explanation: Every finite field has order p^n where p is prime and n is a positive integer. This comes from viewing the field as a vector space over its prime subfield.
The “prime subfield” of any field has size
A p^n always
B 0
C p
D infinite always
Explanation: The smallest subfield generated by 1 is isomorphic to ℤ_p when characteristic is prime p, or to ℚ when characteristic is 0. In prime characteristic, it has p elements.
In a Boolean ring, every element satisfies
A a²=0
B a³=a²
C a+a=a
D a²=a
Explanation: Boolean rings are defined by the identity a²=a for all elements. This forces special behavior: such rings are commutative and closely relate to logic and set operations.
Direct product of groups G×H has operation
A cross-multiplication
B minimum rule
C componentwise
D maximum rule
Explanation: In G×H, (g1,h1)(g2,h2)=(g1g2, h1h2). Each coordinate uses its own group operation. Identity is (e_G,e_H), and inverses are taken coordinatewise too.
A Cayley table is used to show
A operation results
B subgroup index
C element orders only
D ring ideals only
Explanation: A Cayley table lists the product (or sum) of every pair of elements in a finite structure. It helps check closure and can visually confirm properties like identity and inverses.
A permutation group is a group under
A addition
B composition
C set union
D scalar product
Explanation: Permutations are bijections from a set to itself. They form a group under function composition: composition is associative, identity permutation exists, and every permutation has an inverse permutation.