Top Interview Questions on Automorphism Groups

Here are some interview questions that are used on PhD exams, along with suggestions from candidates who have taken them. Hopefully, this page will be helpful for you in your preparation for PhD Interviews & Exams. This piece of information is shared only for educational purposes.

We request all of you, if you have recently appeared in a Ph. D. interview or Assistant Prof. Interview or JRF Position Interview, please share your Interview Que. /Experience /Suggestions to us for others’ benefit, Your little contribution can help New aspirants to prepare themselves.

Editor of this page: P Kalika (went through an interview to get into Shiv Nadar University) (This interview strategy and tips were shared by Dr. Ram Kishor, Professor of mathematics at Central University of Rajasthan. Interview with NBHM for PhD Exam ) (This Interview Experience is Shared by Gautam Kaushik (PhD Aspirant ). ).

❤️ We hearty thanks to those persons who shared their Interview Questions & Experiences for others’ help. We wish you all the best for your future.

I was asked my preference and I replied with Basic Analysis and Algebra. They asked me two questions, one from Analysis and the other from Algebra.

These are the interview questions that were shared by Drishti Sunder Phukon in June 2023 on NBHM, TIFR-CAM, IISER TVM (through the NBHM channel), and NISER.

Sir, I’d like to send interview questions to NISER, NBHM, TIFR-CAM, IISER TVM (through the NBHM channel), and NBHM so that they can help other people. Please find attached the pdf of the same.

I got selected for NBHM masters scholarship, and waiting for results from IISER TVM and NISER.

“I got an email from IIT Ropar last night telling me I got into the PhD program. Your list of questions helped me figure out how to prepare for my interview. Thank you so much from the bottom of my heart.” I have question that they ask during my interview. I am attaching here.

There were PhD interview questions at IIT Bombay, Kanpur, Jodhpur, Hyderabad, and Madras.

‘I am Anik Bhowmick, I have appeared in several PhD interviews this year. I have attached a pdf so that you can share it and help students around the world. ’.

I had given an interview at IIT Guwahati and IISER Pune in the June session. In IISER Pune, they asked questions from linear algebra, ordinary differential equations, real analysis. They are as follows.

Hello ! I attended the PhD interview at IIT Dharwad this year. I’m sharing the questions below.

PhD test and interview questions collection shared by Nidhi Shukla, Download the PDF file (17Pages).

Hello Sir, I am selected for PhD at Aligarh Muslim University. I thank you for all the updates and notes and videos. Sharing my interview questions. Pls keep anonymous. Thank you again for running the kalika community.

Sir, I have appeared for an Interview for VIT Bhopal on 8th december and also got selected. I am attaching the file of questions here. Also big thanks to you sir as the collections of interview questions is very helpful for me. and also I am following your study material of many topics for preparing for CSIR & GATE. Regards Suneel Kumar.

Hey, I have given Interview at UNIVERSITY OF DELHI. Here are the interview questions: For starters, we have to give an 8-minute presentation on any subject we choose. The questions were about “Fundamental Theorem of Finite Abelian Groups.” Here are some of them: 1) What are elementary divisors? Give an example. 2) Given two finite Abelian Groups G and H. How can we use elementary divisors to see if they are isomorphic or not? 4) What is the partial opposite of Lagrange’s theorem for non-Abelian groups that we get from “Fundamental Theorem of Finite Abelian Groups”? 5) How many groups of order 15 are there? 6) What is the Cauchy theorem? 7) How can we use the Sylow theorem to show that the group of order 15 is cyclic? 8) Do you know what a free Abelian group is? 9) What is a commutator group? 10) What are derived series? 11) Can we define a solveable group using derived series? 12) What is a characteristic subgroup? 13) What is the relationship between a characteristic subgroup and a normal subgroup? 14) Why is every characteristic subgroup normal? 15) What is the commutator subgroup of GL_n(R)? Why?.

Hey, I have given an interview at IISER MOHALI. 1. What is the unit group of Z_n? (in terms of n)
2. What are the maximal ideals of Z? (I said of type pZ, where p is prime)
3. How do I prove that the maximal ideals of Z are only of type pZ, where p is prime?
4. Think about the complex polynomial ring C[x]. 5) What are the maximal ideals of the ring C[x,y]? 6) Are the ideals and in the ring C[x] the same? If so, under what conditions for ‘a’ and ‘b’? 7) Is the space L²[a,b] a Hilbert space? 8) What are the maximal ideals of the ring [0,1]? As every point in [0,1] can be written in decimal expansion. Consider the set of all points in [0,1] whose decimal expansion do not include the number 5. What is its measure? .

Me Nowshad Ali Mir from Jammu and Kashmir. I have appeared for interview on 20th of Oct, 2021 at VIT VELLORE. INTERVIEWERS asked me simple questions:

IIT Kanpur, Madrash, BHU & IISER Berhampur PhD Interview Dec 2021 Questions (Anonymous Share)

This is ————, and I was accepted to do a PhD at IIT Madras in January 2022. Here are the interview questions I was asked. Click on the link to read.

1st 10 mins, they show the 1st 5 questions which I had to solve in 10mins. Next few questions they asked verbally.

1) Tell us about your favorite theorem in Linear Algebra. Prove it. 2) What is the Characteristic Polynomial of a matrix? 3) What is the Rank of the matrix? How do we find it?

Interview for the NBHM Ph.D. Scholarship 2021

I have appeared for BITS Pilani Interview and have been selected in it as well. I don’t remember the questions, but there were only three or four of them. One of them was, “What is the Nilpotent Group?” The interview was very simple.

Hello sir recently I experienced NBHM PhD interview and sending my interview questions here. (Shared by Parna Saha).

First they asked my preference and I said abstract algebra. Then they asked two questions from the nbhm phd written test. One is to prove the formula “number of idempotent elements in Zn = 2^d.” The other is from linear algebra, and I got that question wrong on the written exam, but they told me to try again. Then they asked

Advice: If you are going to an NBHM interview, read the proofs of the formula we usually use and study for the NBHM exam. all of them And whatever your preference can be…they ask question from real analysis and topology.

I am attaching here pdf of questions asked in PhD interviews at IIT, IIIT, and IISER. These are collection of questions based on my experience of one year (Shared by Divya).

Hi, this is the question from the PhD interview at the University of Calcutta (Ballygunge Science College) in pure mathematics.

The questions are easy, so when you’re getting ready for the interview, you should be sure you know everything there is to know.

The computer programs we need for the PhD interview are Latex and Matlab. Please try to cover these as soon as possible.

As they ask, “Do you know any computer programs like Latex and Matlab?” I think that is also one of the things that determines who gets chosen and who doesn’t.

This was my first interview. I thought the topic was interesting, so I asked and was told the basics. Maine unhe ode bola, but O bole kisi aur me interested h to maine real bola O bs real se hi Sare question puche aur thodi bahut help bhi krte h

Some of the PhD Interview Questions that were asked to me in the academic year 2020-2021. The interviews were held online (Anonymously Shared).

I selected subjects for interview namely Differential equations and Fuzzy Theory. Because I do project in MPhil on Fuzzy Theory so it’s beneficial for me. They ask some questions about my project.

And from differential equations, they ask what the difference is between SLP and BVP, give an example, and know about boundary conditions. That’s all they ask.

Yes, sir, I think that during an interview, the person should be asked about a certain subject or be told to talk about certain main topics. If that’s not possible, giving examples would be helpful. Sorry, I didn’t mean to use “sorry” or “apni pasand ka topic unhe bta do bss,” but I did. Now that’s all I have to say about it. Thank you sir.

I’ve added it, and I hope the notes are useful. Thank you very much, sir . Isi tarah se hmari help krte rhiye…

Recently I’ve appeared for VIT Vellore and BITS Pilani interview. Here I’m providing my Interview Questions in handwritten format….

” Just prepare some basics … Means whatever you are going to tell about research interests. Prepare basics nicely. Prepare one favourite theorem. Lots of example and counter examples”.

Thank you very much for providing useful tips and updating us on events happening in Mathematical world.

Hello Sir…my suggestions are One may clear the competitive exams but clearing interview is completely different. We need to have good knowledge of topics. Must read standard books (including proofs) like Bartle for Analysis, Gallian for Algebra, etc. Avoid random guess because one wrong answer would expose all your weaknesses. If you don’t know the answer, just say, “I can try,” and if you still can’t get it, say you don’t know the answer. The reasoning behind this must be sound.

Automorphism groups are an important concept in abstract algebra and come up frequently in technical interviews, especially for positions in academia or research. As an interviewee, you need to be prepared to answer common questions about automorphism groups and demonstrate your expertise In this article, I’ll overview some key facts about automorphism groups and provide example answers to common interview questions

What is an Automorphism Group?

An automorphism group is the group of symmetries of a mathematical object that preserve its structure. More formally:

  • Let X be a mathematical object with some structure (such as a group, field, graph, etc.)
  • An automorphism of X is an isomorphism from X to itself that preserves the structure.
  • The automorphism group of X, denoted Aut(X), is the group whose elements are all the automorphisms of X.

For example the automorphism group of the integers Z under addition is the group of translations {f(x) = x+n | n in Z}. This preserves the additive structure of Z.

Some key properties of automorphism groups:

  • Aut(X) always contains the identity automorphism.
  • Automorphisms are invertible, so Aut(X) is a subgroup of the symmetric group Sym(X).
  • For finite objects X, Aut(X) is always finite.
  • Aut(X) measures the amount of symmetry in the structure of X. More symmetry leads to a larger automorphism group.

Automorphism groups are useful for gaining insight into structures in algebra, geometry, graph theory, and more.

Sample Interview Questions and Answers

Here are some common automorphism group questions that may come up in an interview:

Q: How would you compute the automorphism group of a graph?

A: There are a few approaches for computing the automorphism group Aut(G) of a graph G:

  • Find all graph automorphisms by brute force – systematically check all permutations of vertices.
  • Use backtracking or pruning techniques to reduce the search space.
  • Transform the problem into graph isomorphism and use software tools.
  • For special classes of graphs, automorphisms can be characterized theoretically.

In an interview, I would focus on explaining the brute force method clearly and discussing optimizations like backtracking. I may also mention some special cases like Aut(K_n) = S_n for complete graphs.

Q: What is the automorphism group of the field of complex numbers under addition and multiplication?

A: The automorphism group here consists of field automorphisms that preserve + and *. For the complex numbers, this is:

Aut(C, +, *) = {z ↦ z, z ↦ z̅}

The only automorphisms are the identity and complex conjugation. This group is isomorphic to Z/2Z.

Q: How are automorphism groups useful in Galois theory?

A: In Galois theory, one studies field extensions K/F by looking at the group of automorphisms Aut(K/F) that fix F. This is called the Galois group and captures symmetries between roots of a polynomial. Properties of the Galois group like solvability correspond to properties of the extension.

Key uses of automorphism groups in Galois theory:

  • Gal(K/F) controls which elements can be expressed in terms of radicals.
  • Solvability of Gal(K/F) determines if an equation can be solved by radicals.
  • The size of Gal(K/F) determines the degree of a minimal polynomial.

So automorphism groups are fundamentally important for understanding solvability and constructions in field extensions.

Q: How would you explain automorphism towers to a layperson?

A: Here’s how I would explain it:

Think of a robotic arm that is controlled by a computer – the arm can move in different ways without changing what it is. We can think of those movements as “automorphisms” of the arm – symmetries that don’t change its underlying structure.

Now imagine building a robot that tries to control itself, so the computer becomes another robotic arm. This second arm can move the first arm into different configurations. Now we have a “tower” of two robots, each controlling the layer below it.

This stack of progressively more complex robots is like an automorphism tower in algebra. We start with a mathematical object, then consider symmetries of that object, then symmetries of those symmetries, and so on. Studying how high this tower reaches gives insights into the initial structure.

The key idea is layers of control, like a tower of robots controlling robots. This makes the concept intuitive.

Q: Prove that the automorphism group of a cyclic group Z/nZ is isomorphic to Z/nZ* (group of units modulo n).

A: Here is a proof sketch:

Let φ be an automorphism of Z/nZ. Since φ must send generators to generators, φ is determined by where it sends 1 (the generator). φ(1) can be any element relatively prime to n, so there are φ(n) choices.

Conversely, mapping 1 to any number k relatively prime to n gives a valid automorphism. So there is a bijection between Aut(Z/nZ) and Z/nZ*.

It remains to show this bijection is a homomorphism. We observe φ(a) = φ(1+1+…+1) = φ(1) + φ(1) + … + φ(1) = ka for any a. So Aut(Z/nZ) ≅ Z/nZ as groups.

This shows the key ideas: relating automorphisms to units modulo n and showing the bijection respects the group structure.

Q: How can you use the orbit-stabilizer theorem to count automorphisms?

A: The orbit-stabilizer theorem gives a relationship between the size of a group G acting on a set X and the size of orbits/stabilizers. Specifically:

|G| = [G:Stab(x)] * |Orb(x)|

where Stab(x) is the stabilizer of x in X and Orb(x) is the orbit of x.

We can apply this to automorphism groups acting on elements of X. The orbits correspond to how elements of X are permuted. If we pick x in X and know |Stab(x)|, we can use the OS theorem to deduce |Aut(X)|. This is useful for counting automorphisms when stabilizers are easy to determine.

Q: What are some of the open problems around automorphism groups of free groups?

A: Some open questions about Aut(F_n) for a free group F_n include:

  • What is the asymptotic growth rate of Aut(F_n) as n increases? Currently only bounds are known.

  • What is the abelianization of Aut(F_n)? This is known for some small n but the general structure is unclear.

  • How large is the finite subgroup Aut(F_n)/Inn(F_n)? This is bounded but not fully classified.

  • What is the computational complexity of the automorphism problem for F_n? It is known to be in AM but not known to be in NP.

  • Can finite presentations for Aut(F_n) be found? This is open even for n = 3.

This remains an active area of research in combinatorial and geometric group theory. The automorphism groups of free groups exhibit a range of interesting and complex behavior.


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What is an example of automorphism of a group?

Examples. In set theory, an arbitrary permutation of the elements of a set X is an automorphism. The automorphism group of X is also called the symmetric group on X. In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial automorphism: negation.

What’s the difference between isomorphism and automorphism?

Simply, an isomorphism is also called automorphism if both domain and range are equal. If f is an automorphism of group (G,+), then (G,+) is an Abelian group. Identity mapping as we see, in example, is an automorphism over a group is called trivial automorphism and other non-trivial.

What is the difference between automorphism and homomorphism?

A homomorphism that is both injective and surjective is an an isomorphism. An automorphism is an isomorphism from a group to itself. If we know where a homomorphism maps the generators of G, we can determine where it maps all elements of G. For example, suppose φ : Z3 → Z6 was a homomorphism, with φ(1) = 4.

What is the difference between automorphism and inner automorphism?

Automorphism of a group G can be defined as an isomorphism of G onto itself. A trivial case would be an identity map I, such that I(k) = k, for all k ∈ G. I is an automorphism of G. Inner automorphism of G is an automorphism obtained by conjugation by an element.

What is automorphism of a group?

An automorphism of a group is permutation of its elements which preserves the operation, i.e. φ(xy) = φ(x)φ(y) φ ( x y) = φ ( x) φ ( y). Since every group G G is a set, you can look at two possible automorphism groups: one – AutSet(G) Aut S e t ( G) as a group. Cleraly AutGp(G) ≤ AutSet(G) Aut G p ( G), but usually they are not equal.

What is automorphism in set theory?

Looking on the Wikipedia page for automorphism; in the examples it first states that in set theory, the automorphism of a set X X is an arbitrary permutation of the elements of X X, and these form the automorphism group, also known as the symmetric group, on X X.

What is automorphism for a group if it is abelian?

5.f (x)=1/x is automorphism for a group (G,*) if it is Abelian. A set of all the automorphisms ( functions ) of a group, with a composite of functions as binary operations forms a group. Simply, an isomorphism is also called automorphism if both domain and range are equal. If f is an automorphism of group (G,+), then (G,+) is an Abelian group.

How is an automorphism determined?

An automorphism is determined by where it sends the generators. An automorphism ˚must send generators to generators. In particular, if G is cyclic, then it determines apermutationof the set of (all possible) generators. Examples 1.There are two automorphisms of Z: the identity, and the mapping n 7!n. Thus, Aut(Z) ˘=C

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