Algebra – Mathematics Optional PYQs (2008–2026)

Here are all 85 Algebra questions asked in UPSC Mathematics Optional Paper II from 2008 to 2026, arranged under the 5 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

1.1] Groups and subgroups

17 questions

1. Let $H$ and $K$ be two subgroups of a group $G$ such that $o\left(H\right)>\sqrt{o\left(G\right)}$ and $o\left(K\right)>\sqrt{o\left(G\right)}$. Show that $H\cap K\ne \{e\}$, where $e$ is the identity element. Here $o\left(H\right)$, $o\left(K\right)$ and $o\left(G\right)$ denote the order of $H$, $K$ and $G$ respectively. [2025/10m/1a]

2. Let $G=\{e,x,x^2,y,yx,y{x}^2\}$ be a non-Abelian group with $o\left(x\right)=3$ and $o\left(y\right)=2$. Show that $xy=y{x}^2$ (where $e$ is the identity element of $G$ and $o\left(x\right)$, $o\left(y\right)$ denote the order of the elements $x$, $y$ respectively). [2025/10m/1b]

3. Show that there are infinitely many subgroups of the additive group $\mathbb{Q}$ of rational numbers. [2021/15m/4a]

4. Let $G$ be a finite cyclic group of order $n$. Then prove that $G$ has $\varphi \left(n\right)$ generators (where $\varphi$ is Euler’s $\varphi$-function). [2020/15m/2a]

5. Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $K\subset H$. Show that $(G:K)=(G:H)(H:K)$. [2019/10m/1a]

6. Find all the proper subgroups of the multiplicative group of the field $({\mathbb{Z}}_{13},{+}_{13},{\times }_{13})$, where ${+}_{13}$ and ${\times }_{13}$ represent addition modulo 13 and multiplication modulo 13 respectively. [2018/20m/3a]

7. Let $p$ be a prime number and ${\mathbb{Z}}_p$ denote the additive group of integers modulo $p$. Show that every non-zero element of ${\mathbb{Z}}_p$ generates ${\mathbb{Z}}_p$. [2016/15m/2b]

8. (i) How many generators are there of the cyclic group $G$ of order 8? Explain. (ii) Taking a group $\{e,a,b,c\}$ of order 4, where $e$ is the identity, construct composition tables showing that one is cyclic while the other is not. [2015/10m/1a]

9. Give an example of an infinite group in which every element has finite order. [2013/10m/1b]

10. How many elements of order 2 are there in the group of order 16 generated by $a$ and $b$ such that the order of $a$ is 8, the order of $b$ is 2 and $ba{b}^{-1}=a^{-1}$. [2012/12m/1a]

11. Show that the set $G=\{f_1,f_2,f_3,f_4,f_5,f_6\}$ of six transformations on the set of Complex numbers defined by $f_1\left(z\right)=z,f_2\left(z\right)=1-z$ $f_3\left(z\right)=\frac{z}{(z-1)},f_4\left(z\right)=\frac{1}{z}$ $f_5\left(z\right)=\frac{1}{(1-z)}\mathrm{\text{and}}{f}_6\left(z\right)=\frac{(z-1)}{z}$ is a non-abelian group of order 6 w.r.t. composition of mappings. [2011/12m/1a]

12. Prove that a group of Prime order is abelian. [2011/6m/1e(i)]

13. How many generators are there of the cyclic group $(G,\cdot )$ of order 8 ? [2011/6m/1e(ii)]

14. Give an example of a group $G$ in which every proper subgroup is cyclic but the group itself is not cyclic. [2011/15m/2a]

15. Let $a$ and $b$ be elements of a group, with $a^2=e$, $b^6=e$ and $ab=b^{4}a$. Find the order of $ab$, and express its inverse in each of the forms $a^m{b}^n$ and $b^m{a}^n$. [2011/20m/4a]

16. Let $G=ℝ-\{-1\}$ be the set of all real numbers omitting $-1$. Define the binary relation $*$ on $G$ by $a*b=a+b+ab$. Show $(G,*)$ is a group and it is abelian. [2010/12m/1a]

17. Let $R_0$ be the set of all real numbers except zero. Define a binary operation $*$ on $R_0$ as : $a*b=\left|a\right|b$ where $\left|a\right|$ denotes absolute value of $a$. Does $(R_0,*)$ form a group? Examine. [2008/12m/1a]

1.2] Normal subgroups and homomorphisms

17 questions

1. Let $G$ be a group of order $pq$, where $p$ and $q$ are primes. Show that either $G$ is an abelian group or no non-identity element commutes with every element of $G$. [2026/10m/1a]

2. Let $G$ be a finite group generated by two elements $u$ and $v$ of order 2 each. Show that $G$ is isomorphic to the dihedral group of order $2n$, where $O\left(uv\right)=n$. [2026/15m/2a]

3. Show that every homomorphic image of an abelian group is abelian, but the converse is not necessarily true. [2024/15m/2b]

4. Let $G$ be a group of order 10 and $G\prime$ be a group of order 6. Examine whether there exists a homomorphism of $G$ onto $G\prime$. [2023/10m/1a]

5. Show that the multiplicative group $G=\{1,-1,i,-i\}$, where $i=\sqrt{(-1)}$, is isomorphic to the group $G\prime =(\{0,1,2,3\},{+}_4)$. [2022/10m/1a]

6. Prove that every homomorphic image of a group $G$ is isomorphic to some quotient group of $G$. [2022/15m/2b]

7. Let $S_3$ and ${\mathbb{Z}}_3$ be permutation group on 3 symbols and group of residue classes module 3 respectively. Show that there is no homomorphism of $S_3$ in ${\mathbb{Z}}_3$ except the trivial homomorphism. [2020/10m/1a]

8. If $G$ and $H$ are finite groups whose orders are relatively prime, then prove that there is only one homomorphism from $G$ to $H$, the trivial one. [2019/10m/2a]

9. Write down all quotient groups of the group ${\mathbb{Z}}_{12}$. [2019/10m/2b]

10. Show that the quotient group of $(ℝ,+)$ modulo $\mathbb{Z}$ is isomorphic to the multiplicative group of complex numbers on the unit circle in the complex plane. Here $ℝ$ is the set of real numbers and $\mathbb{Z}$ is the set of integers. [2018/15m/2a]

11. Show that the groups ${\mathbb{Z}}_5\times {\mathbb{Z}}_7$ and ${\mathbb{Z}}_{35}$ are isomorphic. [2017/15m/3a]

12. Let $G$ be the set of all real $2\times 2$ matrices $\left[\begin{array}{cc}\hfill x\hfill & \hfill y\hfill \\ \hfill 0\hfill & \hfill z\hfill \end{array}\right]$, where $xz\ne 0$. Show that $G$ is a group under matrix multiplication. Let $N$ denote the subset $\{\left[\begin{array}{cc}\hfill 1\hfill & \hfill a\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]:a\in ℝ\}$. Is $N$ a normal subgroup of $G$? Justify your answer. [2014/10m/1a]

13. Show that a cyclic group of order 6 is isomorphic to the product of a cyclic group of order 2 and a cyclic group of order 3. Can you generalize this ? Justify. [2010/12m/1b]

14. Let $({ℝ}^{*},\cdot )$ be the multiplicative group of non-zero reals and $(GL(n,ℝ),X)$ be the multiplicative group of $n\times n$ non-singular real matrices. Show that the quotient group $GL(n,ℝ)/SL(n,ℝ)$ and $({ℝ}^{*},\cdot )$ are isomorphic where $SL(n,ℝ)=\{A\in GL(n,ℝ)/detA=1\}$. What is the centre of $GL(n,ℝ)$ ? [2010/15m/2a]

15. If $ℝ$ is the set of real numbers and ${ℝ}_{+}$ is the set of positive real numbers, show that $ℝ$ under addition $(ℝ,+)$ and ${ℝ}_{+}$ under multiplication $({ℝ}_{+},\cdot )$ are isomorphic. Similarly if $\mathbb{Q}$ is the set of rational numbers and ${\mathbb{Q}}_{+}$ the set of positive rational numbers, are $(\mathbb{Q},+)$ and $({\mathbb{Q}}_{+},\cdot )$ isomorphic? Justify your answer. [2009/12m/1a]

16. Determine the number of homomorphisms from the additive group ${\mathbb{Z}}_{15}$ to the additive group ${\mathbb{Z}}_{10}$. (${\mathbb{Z}}_n$ is the cyclic group of order $n$). [2009/12m/1b]

17. Let $G$ and $\overline{G}$ be two groups and let $\varphi :G\to \overline{G}$ be a homomorphism. For any element $a\in G$ (i) prove that $O\left(\varphi \left(a\right)\right)/O\left(a\right)$. (ii) $Ker\varphi$ is a normal subgroup of $G$. [2008/15m/2a]

1.3] Permutation groups

7 questions

1. Let $G$ be a finite group of order $mn$, where $m$ and $n$ are prime numbers with $m>n$. Show that $G$ has at most one subgroup of order $m$. [2024/10m/1a]

2. Prove that a non-commutative group of order $2p$, where $p$ is an odd prime, must have a subgroup of order $p$. [2023/15m/2a]

3. Let $G$ be a group of order $n$. Show that $G$ is isomorphic to a subgroup of the permutation group $S_n$. [2017/10m/1b]

4. What are the orders of the following permutations in $S_{10}$ ? $\left(\begin{array}{cccccccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill & \hfill 10\hfill \\ \hfill 1\hfill & \hfill 8\hfill & \hfill 7\hfill & \hfill 3\hfill & \hfill 10\hfill & \hfill 5\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 6\hfill & \hfill 9\hfill \end{array}\right)\mathrm{\text{and}}\left(12345\right)\left(67\right).$ [2013/10m/2a]

5. What is the maximal possible order of an element in $S_{10}$ ? Why ? Give an example of such an element. How many elements will there be in $S_{10}$ of that order ? [2013/13m/2b]

6. How many conjugacy classes does the permutation group $S_5$ of permutations 5 numbers have ? Write down one element in each class (preferably in terms of cycles). [2012/15m/2a]

7. Show that the alternating group on four letters $A_4$ has no subgroup of order 6. [2009/15m/2b]

1.4] Rings and ideals

16 questions

1. Examine whether the mapping $\varphi :\mathbb{Z}[x]\to \mathbb{Z}$ defined by $\varphi \left(f\left(x\right)\right)=f\left(0\right)$, for $f\left(x\right)\in \mathbb{Z}[x]$, is a homomorphism. Deduce that the ideal $⟨x⟩$ is a prime ideal in $\mathbb{Z}[x]$, but not a maximal ideal in $\mathbb{Z}[x]$. [2025/15m/4a]

2. Consider the polynomial ring $\mathbb{Z}[x]$ over the ring $\mathbb{Z}$ of integers. Let $S$ be an ideal of $\mathbb{Z}[x]$ generated by $x$. Show that $S$ is prime but not a maximal ideal of $\mathbb{Z}[x]$. [2024/15m/4a]

3. Let $ℝ$ be a field of real numbers and $S$, the field of all those polynomials $f\left(x\right)\in ℝ[x]$ such that $f\left(0\right)=0=f\left(1\right)$. Prove that $S$ is an ideal of $ℝ[x]$. Is the residue class ring $ℝ[x]/S$ an integral domain? Give justification for your answer. [2022/15m/4a]

4. Suppose $ℝ$ be the set of all real numbers and $f:ℝ\to ℝ$ is a function such that the following equations hold for all $x,y\in ℝ$ : (i) $f(x+y)=f\left(x\right)+f\left(y\right)$ (ii) $f\left(xy\right)=f\left(x\right)f\left(y\right)$ Show that $\forall x\in ℝ$ either $f\left(x\right)=0$, or, $f\left(x\right)=x$. [2018/20m/4a]

5. Let $𝕂$ be a field and $𝕂[X]$ be the ring of polynomials over $𝕂$ in a single variable $X$. For a polynomial $f\in 𝕂[X]$, let $\left(f\right)$ denote the ideal in $𝕂[X]$ generated by $f$. Show that $\left(f\right)$ is a maximal ideal in $𝕂[X]$ if and only if $f$ is an irreducible polynomial over $𝕂$. [2016/10m/1a]

6. Give an example of a ring having identity but a subring of this having a different identity. [2015/10m/1b]

7. If $R$ is a ring with unit element 1 and $\varphi$ is a homomorphism of $R$ onto $R\prime$, prove that $\varphi \left(1\right)$ is the unit element of $R\prime$. [2015/15m/2a]

8. Prove that the set $\mathbb{Q}\left(\sqrt{5}\right)=\{a+b\sqrt{5}:a,b\in \mathbb{Q}\}$ is a commutative ring with identity. [2014/15m/4a]

9. Let $R^C$ = ring of all real valued continuous functions on $[0,1]$, under the operations $\begin{array}{cc}\hfill (f+g)x\hfill & \hfill =f\left(x\right)+g\left(x\right)\hfill \\ \hfill \left(fg\right)x\hfill & \hfill =f\left(x\right)g\left(x\right).\hfill \end{array}$ Let $M=\{f\in R^C\mid f\left(\frac{1}{2}\right)=0\}$. Is $M$ a maximal ideal of $R$ ? Justify your answer. [2013/15m/3b]

10. Describe the maximal ideals in the ring of Gaussian integers $\mathbb{Z}[i]=\{a+bi\mid a,b\in \mathbb{Z}\}$. [2012/20m/4a]

11. Let $F$ be the set of all real valued continuous functions defined on the closed interval $[0,1]$. Prove that $(F,+,\cdot )$ is a Commutative Ring with unity with respect to addition and multiplication of functions defined pointwise as below : $\begin{array}{cc}\hfill \hfill & \hfill (f+g)\left(x\right)=f\left(x\right)+g\left(x\right)\hfill \\ \hfill \mathrm{\text{and}}\hfill & \hfill (f\cdot g)\left(x\right)=f\left(x\right)\cdot g\left(x\right)\hfill \end{array}\}x\in [0,1]$ where $f,g\in F$. [2011/15m/3a]

12. Show that the quotient ring $\mathbb{Z}[i]/(1+3i)$ is isomorphic to the ring $\mathbb{Z}/10\mathbb{Z}$ where $\mathbb{Z}[i]$ denotes the ring of Gaussian integers. [2010/15m/3b]

13. How many proper, non-zero ideals does the ring ${\mathbb{Z}}_{12}$ have ? Justify your answer. How many ideals does the ring ${\mathbb{Z}}_{12}\oplus {\mathbb{Z}}_{12}$ have ? Why ? [2009/15m/2a]

14. How many elements does the quotient ring $\frac{{\mathbb{Z}}_5[X]}{(X^2+1)}$ have ? Is it an integral domain ? Justify yours answers. [2009/15m/3b]

15. Suppose that there is a positive even integer $n$ such that $a^n=a$ for all the elements ‘$a$‘ of some ring $R$. show that $a+a=0$ for all $a\in r$ and $a+b=0\Rightarrow a=b$ for all $a,b\in R$. [2008/12m/1b]

16. Show that any maximal ideal in the commutative ring $F[x]$ of polynomials over a field $f$ is the principal ideal generated by an irreducible polynomial. [2008/15m/3b]

1.5] Integral domains and fields

28 questions

1. Show that every non-zero prime ideal in a Euclidean Domain is maximal. [2026/10m/1b]

2. Let $n$ be a positive integer. Show that the following statements are equivalent : (i) $\mathbb{Z}/n\mathbb{Z}$ is an integral domain. (ii) $\mathbb{Z}/n\mathbb{Z}$ is a field. (iii) $n$ is prime. [2026/15m/4a]

3. Show that 3 is an irreducible element in the integral domain $\mathbb{Z}[i]$. [2025/15m/2b]

4. Express the ideal $4\mathbb{Z}+6\mathbb{Z}$ as a principal ideal in the integral domain $\mathbb{Z}$. [2023/10m/1b]

5. Prove that $x^2+1$ is an irreducible polynomial in ${\mathbb{Z}}_3[x]$. Further show that the quotient ring $\frac{{\mathbb{Z}}_3[x]}{⟨x^2+1⟩}$ is a field of 9 elements. [2023/15m/3a]

6. Let $m_1,m_2,\cdots ,m_k$ be positive integers and $d>0$ the greatest common divisor of $m_1,m_2,\cdots ,m_k$. Show that there exist integers $x_1,x_2,\cdots ,x_k$ such that $d=x_1{m}_1+x_2{m}_2+\cdots +x_k{m}_k$ [2021/10m/1a]

7. Let $F$ be a field and $f\left(x\right)\in F[x]$ a polynomial of degree $>0$ over $F$. Show that there is a field $F\prime$ and an imbedding $q:F\to F\prime$ s.t. the polynomial $f^q\in F\prime [x]$ has a root in $F\prime$, where $f^q$ is obtained by replacing each coefficient $a$ of $f$ by $q\left(a\right)$. [2021/15m/2b]

8. Let $R$ be a principal ideal domain. Show that every ideal of a quotient ring of $R$ is principal ideal and $R/P$ is a principal ideal domain for a prime ideal $P$ of $R$. [2020/10m/1b]

9. Let $R$ be a finite field of characteristic $p(>0)$. Show that the mapping $f:R\to R$ defined by $f\left(a\right)=a^p$, $\forall a\in R$ is an isomorphism. [2020/15m/3a]

10. Let $a$ be an irreducible element of the Euclidean ring $R$, then prove that $R/\left(a\right)$ is a field. [2019/10m/3d]

11. Let $R$ be an integral domain with unit element. Show that any unit in $R[x]$ is a unit in $R$. [2018/10m/1a]

12. Let $F$ be a field and $F[X]$ denote the ring of polynomials over $F$ in a single variable $X$. For $f\left(X\right),g\left(X\right)\in F[X]$ with $g\left(X\right)\ne 0$, show that there exist $q\left(X\right),r\left(X\right)\in F[X]$ such that $\mathrm{\text{degree}}\left(r\left(X\right)\right)<\mathrm{\text{degree}}\left(g\left(X\right)\right)$ and $f\left(X\right)=q\left(X\right)\cdot g\left(X\right)+r\left(X\right).$ [2017/20m/2c]

13. Let $𝕂$ be a field and $𝕂[X]$ be the ring of polynomials over $𝕂$ in a single variable $X$. For a polynomial $f\in 𝕂[X]$, let $\left(f\right)$ denote the ideal in $𝕂[X]$ generated by $f$. Show that $\left(f\right)$ is a maximal ideal in $𝕂[X]$ if and only if $f$ is an irreducible polynomial over $𝕂$. [2016/10m/1a]

14. Let $K$ be an extension of a field $F$. Prove that the elements of $K$, which are algebraic over $F$, form a subfield of $K$. Further, if $F\subset K\subset L$ are fields, $L$ is algebraic over $K$ and $K$ is algebraic over $F$, then prove that $L$ is algebraic over $F$. [2016/20m/3a]

15. Show that every algebraically closed field is infinite. [2016/15m/4a]

16. Do the following sets form integral domains with respect to ordinary addition and multiplication? If so, state if they are fields : (i) The set of numbers of the form $b\sqrt{2}$ with $b$ rational (ii) The set of even integers (iii) The set of positive integers. [2015/15m/4a]

17. Show that ${\mathbb{Z}}_7$ is a field. Then find ${([5]+[6])}^{-1}$ and ${(-[4])}^{-1}$ in ${\mathbb{Z}}_7$. [2014/15m/2a]

18. Show that the set $\{a+b\omega :{\omega }^3=1\}$, where $a$ and $b$ are real numbers, is a field with respect to usual addition and multiplication. [2014/15m/3a]

19. Show that the set of matrices $S=\{\left(\begin{array}{cc}\hfill a\hfill & \hfill -b\hfill \\ \hfill b\hfill & \hfill a\hfill \end{array}\right)\mid a,b\in ℝ\}$ is a field under the usual binary operations of matrix addition and matrix multiplication. What are the additive and multiplicative identities and what is the inverse of $\left(\begin{array}{cc}\hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$ ? Consider the map $f:ℂ\to S$ defined by $f(a+ib)=\left(\begin{array}{cc}\hfill a\hfill & \hfill -b\hfill \\ \hfill b\hfill & \hfill a\hfill \end{array}\right)$. Show that $f$ is an isomorphism. (Here $ℝ$ is the set of real numbers and $ℂ$ is the set of complex numbers.) [2013/10m/1a]

20. Let $J=\{a+bi\mid a,b\in \mathbb{Z}\}$ be the ring of Gaussian integers (subring of $ℂ$). Which of the following is $J$ : Euclidean domain, principal ideal domain, unique factorization domain ? Justify your answer. [2013/15m/3a]

21. Is the ideal generated by 2 and $X$ in the polynomial ring $\mathbb{Z}[X]$ of polynomials in a single variable $X$ with coefficients in the ring of integers $\mathbb{Z}$, a principal ideal ? Justify your answer. [2012/15m/3a]

22. Let $C=\{f:I=[0,1]\to ℝ\mid f\mathrm{\text{is continuous}}\}$. Show $C$ is a commutative ring with 1 under pointwise addition and multiplication. Determine whether $C$ is an integral domain. Explain. [2010/15m/2b]

23. Consider the polynomial ring $Q[x]$. Show $p\left(x\right)=x^3-2$ is irreducible over $Q$. Let $I$ be the ideal in $Q[x]$ generated by $p\left(x\right)$. Then show that $Q[x]/I$ is a field and that each element of it is of the form $a_0+a_{1}t+a_2{t}^2$ with $a_0$, $a_1$, $a_2$ in $Q$ and $t=x+I$. [2010/15m/3a]

24. Show that $\mathbb{Z}[X]$ is a unique factorization domain that is not a principal ideal domain ($\mathbb{Z}$ is the ring of integers). Is it possible to give an example of principal ideal domain that is not a unique factorization domain ? ($\mathbb{Z}[X]$ is the ring of polynomials in the variable $X$ with integer.) [2009/15m/3a]

25. How many elements does the quotient ring $\frac{{\mathbb{Z}}_5[X]}{(X^2+1)}$ have ? Is it an integral domain ? Justify yours answers. [2009/15m/3b]

26. Let $R$ be a ring with unity. If the product of any two non zero elements is non zero prove that $ab=1\Rightarrow ba=1$. Whether ${\mathbb{Z}}_6$ has the above property or not explain. If ${\mathbb{Z}}_6$ in integral domain? [2008/15m/2b]

27. Prove that every Integral Domain can be embedded in a field. [2008/15m/3a]

28. Show that any maximal ideal in the commutative ring $F[x]$ of polynomials over a field $f$ is the principal ideal generated by an irreducible polynomial. [2008/15m/3b]

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Last updated: 30 October 2026