GATE Mathematics Algebra Syllabus

Algebra syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Algebra in GATE Mathematics, not a summary of it.

1. 1

   Groups

   8 topics

   • Subgroups
   • Normal Subgroups
   • Quotient Groups
   • Homomorphisms
   • Cyclic Groups
   • Permutation Groups
   • Group Action
   • Sylow’s Theorems and Their Applications
2. 2

   Rings

   8 topics

   • Ideals
   • Prime and Maximal Ideals
   • Quotient Rings
   • Unique Factorization Domains
   • Principle Ideal Domains
   • Euclidean Domains
   • Polynomial Rings
   • Eisenstein’s Irreducibility Criterion
3. 3

   Fields

   4 topics

   • Finite Fields
   • Field Extensions
   • Algebraic Extensions
   • Algebraically Closed Fields

Algebra flashcards for GATE Mathematics

19 of 57 cards from the Algebra deck — real questions with worked answers.

1. State the subgroup criterion: when is a nonempty subset $H$ of a group $G$ a subgroup?

   A nonempty subset $H \subseteq G$ is a subgroup iff for all $a,b \in H$ we have $ab^{-1} \in H$ (the one-step subgroup test). Equivalently, $H$ is closed under the operation and under inverses.

2. State Lagrange's Theorem and one immediate consequence for element orders.

   If $G$ is a finite group and $H \leq G$, then $|H|$ divides $|G|$, and $[G:H] = \frac{|G|}{|H|}$. Consequence: the order of every element $g \in G$ divides $|G|$, so $g^{|G|} = e$.

3. For a subgroup $H \leq G$, how are the left coset $gH$ and the index defined, and what is true of distinct cosets?

   $gH = \{gh : h \in H\}$. The index $[G:H]$ is the number of distinct left cosets. Distinct cosets are disjoint and all have the same cardinality $|H|$, so they partition $G$.

4. Give the definition of a normal subgroup $N \trianglelefteq G$ and two equivalent conditions.

   $N \trianglelefteq G$ means $gNg^{-1} = N$ for all $g \in G$. Equivalent: $gN = Ng$ for all $g$ (left cosets equal right cosets), or $gng^{-1} \in N$ for all $g \in G, n \in N$.

5. Why is every subgroup of index $2$ automatically normal?

   If $[G:H]=2$, there are exactly two left cosets $H$ and $G\setminus H$, and two right cosets $H$ and $G\setminus H$. So $gH = Hg$ for every $g$, hence $H \trianglelefteq G$.

6. Define the kernel and center of a group and state their normality.

   For a homomorphism $\varphi: G \to H$, $\ker\varphi = \{g : \varphi(g)=e_H\}$, which is always normal in $G$. The center $Z(G) = \{z : zg = gz \ \forall g \in G\}$ is also a normal (in fact abelian) subgroup.

7. How is the quotient group $G/N$ defined and what is its order when $G$ is finite?

   For $N \trianglelefteq G$, $G/N$ is the set of cosets $\{gN\}$ with operation $(aN)(bN) = (ab)N$. Its order is $|G/N| = [G:N] = \frac{|G|}{|N|}$.

8. State the First Isomorphism Theorem for groups.

   If $\varphi: G \to H$ is a group homomorphism, then $G/\ker\varphi \cong \operatorname{im}\varphi$. In particular the image is isomorphic to the quotient by the kernel.

9. State the Correspondence (Fourth) Isomorphism Theorem for a normal subgroup $N \trianglelefteq G$.

   There is an inclusion-preserving bijection between subgroups of $G$ containing $N$ and subgroups of $G/N$, given by $K \mapsto K/N$. Normal subgroups correspond to normal subgroups.

10. What is a group homomorphism, and what are its two basic preservation properties?

    A map $\varphi: G \to H$ with $\varphi(ab) = \varphi(a)\varphi(b)$ for all $a,b$. It satisfies $\varphi(e_G) = e_H$ and $\varphi(g^{-1}) = \varphi(g)^{-1}$.

11. When is a homomorphism $\varphi: G \to H$ injective, in terms of its kernel?

    $\varphi$ is injective (a monomorphism) iff $\ker\varphi = \{e_G\}$, the trivial subgroup.

12. State the relationship $|\varphi(g)|$ and $|g|$ for a homomorphism, and the divisibility fact.

    For a homomorphism $\varphi$, the order $|\varphi(g)|$ divides $|g|$ (when $g$ has finite order). If $\varphi$ is injective then $|\varphi(g)| = |g|$.

13. Classify all groups of order $p$ (prime). Why?

    Every group of prime order $p$ is cyclic, $\cong \mathbb{Z}/p\mathbb{Z}$. Any non-identity element has order dividing $p$ (Lagrange), hence order $p$, so it generates the whole group.

14. For a cyclic group $\langle a \rangle$ of order $n$, what is the order of the element $a^{k}$?

    $|a^{k}| = \dfrac{n}{\gcd(n,k)}$. In particular $a^{k}$ generates the group iff $\gcd(n,k) = 1$.

15. How many subgroups does a cyclic group of order $n$ have, and what are their orders?

    For each divisor $d \mid n$ there is exactly one subgroup of order $d$, and these are all the subgroups. The number of subgroups equals the number of divisors $\tau(n)$.

16. How many generators does a cyclic group of order $n$ have?

    It has $\varphi(n)$ generators, where $\varphi$ is Euler's totient function: the number of integers in $\{1,\dots,n\}$ coprime to $n$.

17. Define a transposition and the sign of a permutation. What is $\operatorname{sgn}$ as a homomorphism?

    A transposition is a $2$-cycle $(a\,b)$. The sign $\operatorname{sgn}(\sigma) = (-1)^{m}$ where $\sigma$ is a product of $m$ transpositions (parity is well-defined). $\operatorname{sgn}: S_n \to \{\pm 1\}$ is a homomorphism with kernel $A_n$.

18. What is the order of a permutation given its disjoint cycle decomposition?

    The order equals the least common multiple of the lengths of its disjoint cycles. E.g. a product of a $3$-cycle and a $2$-cycle has order $\operatorname{lcm}(3,2) = 6$.

19. State the orders of $S_n$ and $A_n$, and the simplicity fact for $A_n$.

    $|S_n| = n!$ and $|A_n| = \frac{n!}{2}$ for $n \geq 2$. The alternating group $A_n$ is simple for all $n \geq 5$.

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