Should I be worried that I am doing well in analysis and not well in algebra? [closed]

I believe that I may be of some consolation.

I had a very similar experience to you. I started doing "serious" math when I was a senior in high school. I thought I was very smart because I was studying what I thought was advanced analysis--baby Rudin. My ego took a hit when I reached college and realized that while I had a knack for analysis and point-set topology, I could not get this algebra thing down! I just didn't understand what all these sets and maps had to do with anything. I didn't understand why they were useful, and even when I finally did grasp a concept I was entirely impotent when it came to those numbered terrors at the end of chapters.

I held the same fear that you do. I convinced myself that I was destined to be an analyst--I even went as far to say that I "hated" algebra (obnoxious, I know). After about a year of so, with the osmotic effect of being in algebra related classes, and studying tangentially related subjects, I started to understand, and really pick up on algebra. Two years after that (now) I would firmly place myself on the algebraic side of the bridge (if there is such a thing), even though I still enjoy me some analysis!

I think the key for me was picking up the goals and methods of algebra. It is much easier for a gifted math student to "get" analysis straight out of high-school, you have been secretly doing it for years. For the first half of Rudin while I "got it", this was largely thanks to the ability to rely on my calculus background to get why and how we roughly approached things. There was no such helpful intuition for algebra. It was the first type of math I seriously attempted to learn that was "structural", which was qualitative vs. quantitative. My analytic (read calculus) mind was not able to understand why it would ever be obvious to pass from ring X to its quotient, nor why we care that every finitely generated abelian group is a finite product of cyclic groups. I just didn't understand.

But, as I said, as I progressed through more and more courses, learned more and more algebra and related subjects, things just started to click. I not only was able to understand the technical reasons why an exact sequence split, but I understood what this really means intuitively. I started forcing myself to start phrasing other parts of mathematics algebraically, to help my understanding.

The last thing I will say to you, is that you should be scared and worried. I can't tell you how many times in my mathematical schooling I was terrified of a subject. I always thought that I would never understand Subject X or that Concept Y was just beyond me. I can tell you, with the utmost sincerity, that those subjects I was once mortified by, are the subjects I know best. The key is to take your fear that you can't do it, that algebra is just "not your thing", and own it. Be intrigued by this subject you can't understand, read everything you can about it, talk to those who are now good at the subject (even though many of them may have had similar issues), and sooner than you know, by sheer force of will you will find yourself studying topics whose name would make you-right-now die of fright. Stay strong friend, you can do it.

Some people are just naturally more analytic than algebraic, and vice versa. Personally, I do research level algebra, but if I see $\epsilon$ and $\delta$ on the same page I run screaming.

That's not good though, so I'm making myself do it. I enrolled in a complex analysis class, and awful as y'all's side of the fence is, I'm sticking to it. And though I'm not doing the best in the class, y'know, I'm starting to enjoy parts of it.

You can't expect to excel in every area as a mathematician, so focus on rocking at the stuff you do like to do, but make sure your head stays above water in the areas you don't like. You never know when you might end up needing algebra to accomplish something in analysis. If you need motivation, try thinking about hybrid disciplines like functional analysis / operator theory (there's even an "algebraic analysis"). Relate theorems you learn in algebra back to analysis in any way you can. It will help you remember them and maybe give you some cool ideas for later.

No reason to be alarmed or worried...it's too early for you to be in a position to worry about it. For a first crack at abstract algebra, don't fret. Most undergraduate math majors inevitably do fall into "analyis-oriented" and "algebra-oriented", just as in highschool, there is often a "partition" of students into those who prefer high school algebra and those who prefer geometry.

But, it takes more than one course to know this about yourself. Abstract algebra, when I took it as an undergraduate, was typically the gateway course into higher-level abstract math. Personally, I loved it! But there were also classes I developed a love for after covering the "tools" and language of the field: i.e., only after having taken a class or two.

If it's any consolation, you are encountering abstract algebra at a very young age, and though you no doubt have mathematical talent, it does take "time" and effort, and not just raw talent, to develop the cognitive and mathematical maturity to reason abstractly and to develop a sense of "grasping a subject intuitively." Sometimes this is facilitate by classes like an "Introduction to the language and practice of mathematics" (which a Univ near me offers as a bridge between calculus/differential equations and introductory linear algebra, and all subsequent course offerings.

At any rate, wrt becoming comfortable with the more abstract nature of what you're encountering: It is largely a matter of exposure to and engagement with abstract math to operate comfortably within that realm, but there is also a purely developmental component which impacts the ease with which this "acclimation" occurs.

Is this a sign that I simply don't have what it takes to succeed in math?

No, it is not. Every math student I know, sooner or later, has encountered a point where they ask themselves that very question. Often times, more than once.

How you respond at points like this, and how you respond when you feel overwhelmed or intimidated (and you will feel that way again, if you persist in your studies!): that will determine whether you have what it takes to succeed in math.

I am also an undergrad student. I suggest you to calm down. I know I will do a math PhD, so I work hard to enjoy the math and feel no pressure. Sometimes or often I have difficulty in some areas that might be trivial to many people, but I don't feel the rush because I know one day I will understand them, simply by going through them over and over again.

I often compare studying math to playing a computer game with infinitely many levels that gets exponentially harder. Sure, it's good if you make lots of progress and go through many levels fast, but sometimes the point of playing the game is to enjoy the moment and not to always aim at next level while you are playing your current level, because sooner or later you will get stuck at some level and might not go to next for a long long time.

So relax and study and enjoy.