Self-Assessment Quiz: Algebraic Structures — Groups, Rings, and Fields

Twenty questions to check your understanding. Answer from memory before opening the key. Aim for 16+.

Question 1

Which of the following is not one of the four group axioms?

A) closure B) associativity C) commutativity D) existence of inverses

Question 2

The order of a group $G$ means:

A) the smallest $k>0$ with $g^k = e$ for every $g$ B) the number of elements $\lvert G\rvert$ C) the number of generators of $G$ D) the number of subgroups of $G$

Question 3

$(\mathbb{Z}_n, \cdot)$ — all of $\mathbb{Z}_n$ under multiplication — fails to be a group because:

A) multiplication mod $n$ is not associative B) there is no multiplicative identity C) $0$ (and, when $n$ is composite, other elements) has no multiplicative inverse D) it is not closed

Question 4

The set $\mathbb{Z}_n^*$ consists of:

A) all nonzero residues mod $n$ B) the residues $a$ with $1 \le a \le n-1$ and $\gcd(a,n)=1$ C) the residues that are prime D) all residues mod $n$ except $1$

Question 5

$\lvert \mathbb{Z}_n^*\rvert$ equals:

A) $n$ B) $n-1$ C) $\phi(n)$ D) $\phi(n-1)$

Question 6

A subgroup $H$ of $(G, \ast)$ must satisfy all of the following except:

A) $H$ contains the identity $e$ B) $H$ is closed under $\ast$ C) $H$ is closed under taking inverses D) $H$ contains a generator of $G$

Question 7

Lagrange's theorem states that for a finite group $G$ and subgroup $H$:

A) $\lvert G\rvert$ divides $\lvert H\rvert$ B) $\lvert H\rvert$ divides $\lvert G\rvert$ C) $\lvert H\rvert = \lvert G\rvert / 2$ D) $\lvert H\rvert$ and $\lvert G\rvert$ are coprime

Question 8

A group of prime order $p$ has subgroups of which orders?

A) every divisor of $p$ B) only $1$ and $p$ C) only $p$ D) $1, 2, \dots, p$

Question 9

Corollary 24.3 says that in a finite group $G$ of order $N$, every element $a$ satisfies:

A) $a^N = a$ B) $Na = e$ C) $a^N = e$ D) $a = e$

Question 10

Euler's theorem ($a^{\phi(n)} \equiv 1 \pmod n$ for $\gcd(a,n)=1$) is, in this chapter's language:

A) an axiom of $\mathbb{Z}_n$ B) Corollary 24.3 applied to the group $\mathbb{Z}_n^*$ C) the definition of a field D) a special case of Lagrange's theorem with $H = \{e\}$

Question 11

A cyclic group is one in which:

A) every element is its own inverse B) there is a generator $g$ such that every element is a power of $g$ C) the operation is commutative D) the order is prime

Question 12

On a 12-hour clock $(\mathbb{Z}_{12}, +)$, stepping by $3$ repeatedly:

A) visits all twelve numbers B) visits only $\{0, 3, 6, 9\}$ — a subgroup of order 4 C) visits only $\{3\}$ D) never returns to $0$

Question 13

A ring differs from a field in that a ring does not require:

A) an additive identity $0$ B) associativity of multiplication C) that every nonzero element have a multiplicative inverse D) the distributive law

Question 14

$\mathbb{Z}_n$ is a field if and only if:

A) $n$ is even B) $n$ is prime C) $n$ is a prime power D) $n > 1$

Question 15

A finite field $\mathrm{GF}(q)$ exists if and only if:

A) $q$ is prime B) $q$ is even C) $q = p^k$ is a prime power D) $q$ is any integer $\ge 2$

Question 16

In $\mathrm{GF}(2^n)$, addition of two elements is:

A) integer addition mod $2^n$ B) bitwise XOR (coefficient-wise addition mod 2) C) polynomial multiplication D) bitwise AND

Question 17

In the construction of $\mathrm{GF}(2^3)$, the role played by the irreducible polynomial $m(x)$ is analogous to the role played in $\mathbb{Z}_p$ by:

A) the identity $1$ B) the prime $p$ C) the generator D) the number $0$

Question 18

By Theorem 24.6, the nonzero elements of a finite field $\mathrm{GF}(q)$ form:

A) a ring with zero divisors B) a cyclic group of order $q-1$ under multiplication C) a field of order $q-1$ D) a non-abelian group

Question 19 (True/False, justify)

True or false: $\mathrm{GF}(8)$ and $\mathbb{Z}_8$ are the same algebraic structure. Justify in one sentence.

Question 20 (True/False, justify)

True or false: In $\mathrm{GF}(2^n)$, subtraction is the same operation as addition. Justify in one sentence.

Question 21 (Short answer)

State, in the language of group structure, the one-line reason RSA decryption inverts RSA encryption (name the group and the result used).

Question 22 (Short answer)

Give one nonzero element of $\mathbb{Z}_6$ that has no multiplicative inverse, and the zero-divisor equation that explains why.

Question 23 (Short answer)

List the two trivial subgroups that every group $G$ has.

Question 24 (Multiple choice)

Why does coding theory (e.g. Reed–Solomon) require a field rather than just a ring?

A) fields are smaller and faster B) only fields allow division, which the decoding algorithm needs, and fields have no zero divisors C) rings are not associative D) fields have more elements

Question 25 (True/False, justify)

True or false: Commutativity of the operation is required for a set-with-operation to be a group. Justify in one sentence.

Answer Key

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