HPAS Maths Optional PYQs: Modern Algebra

On that page, you will find year-wise and question-wise solutions.

The following is a compilation of all questions pertaining to Modern Algebra (Abstract Algebra), primarily focused on Group Theory, drawn from the provided HPAS Maths Optional Question Papers (2014–2024), presented in reverse chronological order as requested.

HPAS 2024 Modern Algebra Questions

Question 2(a): Let G be a group. Let Aut(G) denote the set of all automorphisms of G and let $A(G)$ be the group of all permutations of G. Show that Aut(G) is a subgroup of $A(G)$.

HPAS 2023 Modern Algebra Questions

Question 1(a): Suppose G is a finite group of order $pq$, where $p$ and $q$ are prime numbers such that $p>q$. Show that G has at most one subgroup of order $p$.

Question 2(a): Show that every finitely generated subgroup of $\langle \mathbb{Q}, +\rangle$ is cyclic, where $\mathbb{Q}$ is the set of rational numbers.

Question 2(b): Show that a subgroup of an infinite cyclic group is infinite.

Question 2(c): Give an example of an infinite group in which every element is of finite order. Justify your answer.

HPAS 2021 Modern Algebra Questions

Question 1(c): Give an example of a finite abelian group which is not cyclic.

Question 2(a): Show that a finite group having more than two elements has a non-trivial automorphism.

Question 2(b): Prove that every quotient group of a cyclic group is cyclic. Does the converse of this statement hold? Justify your answer with an example.

HPAS 2020 Modern Algebra Questions

Question 1(b): Determine all group homomorphisms from $S_3$ to $\mathbb{Z}_3$.

Question 2(a): Prove that any subgroup of a cyclic group is cyclic.

Question 2(b): Define the order of an element in a group. Let G be a finite group of even order. Show that G has an element of order 2 and that the number of elements of order 2 in G is odd.

HPAS 2019 Modern Algebra Questions

Question 3(a): Show that a cyclic group is necessarily abelian. Show by an example that the converse may not be true.

Question 3(b): Examine if there exists a one-to-one correspondence between the right and left cosets of H in G, if H is any subgroup of G.

Question 3(c): If $f:G \to G’$ is a homomorphism, then prove that $\text{Im}(f)$ is a subgroup of $G’$.

HPAS 2018 Modern Algebra Questions

Question 1(a): Let X be any non-empty set and $S(X)$ be the set of all bijections of X onto itself. Then prove that $(S(X), \circ)$ is an abelian group if and only if X is a set with one or two elements, where $\circ$ is the operation of composition of functions.

Question 2(a): Let G be a group that has two subgroups of orders 45 and 75. If $|G| < 400$, find $|G|$.

Question 2(b): Let $f:G \to H$ be a group homomorphism with kernel K. If the orders of G, H, and K are 75, 45, and 15 respectively, find the order of the image $f(G)$.

HPAS 2017 Modern Algebra Questions

Question 1(a): If $a \ne e$ is the only element of order 2 in a group G, then prove that $ax = xa$ for all $x \in G$.

Question 2(a): Prove that a finite group of prime order does not have any proper subgroup.

Question 2(b): Prove that the kernel of a homomorphism $f$ from a group G to a group G’ is a normal subgroup of G.

HPAS 2016 Modern Algebra Questions

Question 1(d): Let G be a group of all $2\times2$ non-singular matrices with real entries. Determine the centre of G.

Question 3(a): Show that the converse of Lagrange’s theorem holds in a finite cyclic group.

Question 3(b): Give an example of two subgroups H and K which are not normal, but HK is a subgroup.

HPAS 2015 Modern Algebra Questions

Question 1(a): Show that the set: $G=\{0,1,2,3,4\}$ is a finite group of order 5 with respect to addition modulo 5.

Question 2(a): Prove that the order of an element of a group is always equal to the order of its inverse.

Question 2(b): Prove that every homomorphic image of group G is isomorphic to some quotient group of G.

HPAS 2014 Modern Algebra Questions

Question 1(a): If $$ A=\begin{pmatrix} 1&2&3&4&5\\ 2&3&1&5&4 \end{pmatrix} $$ is a permutation on five symbols, then find $A^{3}$ and order of A.

Question 2(a): State and prove Cayley’s Theorem.

Question 2(b): Let $I$ be the additive group of Integers. Let $H$ be the subgroup of $I$ such that: $H=\{mx:x\in I\}$. Write the element of the quotient group $\frac{I}{H}$. Also prepare a composition table for $\frac{I}{H}$ when $m=5$.