Inculcation: the many faces of groups

Finite groups §

little things: groups (group theory, group actions) §

• One should start their journey with groups!
• Group theory, abstraction, and the 196,883-dimensional monster - YouTube
• Essence of Group Theory - YouTube
• Chapter 1 and 3 from evan-chen-napkin
• Artin’s Algebra is a good text.

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representation theory of finite groups §

There are different levels to study representation theory of finite groups:

• before doing finite groups when one has basic ideas of matrices/linear algebra one may try to ponder on
  • consider a finite set of $n\times n$ matrices which are closed under multiplication and has inverses:
  • easy example is for an invertible matrix $A$ consider the set

$$\{I,A,A^2,\dots \}$$

this set might be finite or infinite, if this is finite then $A^n=I$ for some $n$, then what can we say about the matrix $A$ - Try to prove that if the field is $\mathbb{C}$ (or any algebraically closed field) $A$ must be diagonalizable. - Find what the eigenvalues of $A$ may be if the field is $\text{C}$. -

representation theory is just spicy linear algebra §

A group homomorphism from a (say, finite for now) group $G$ to the general linear group on a vector space $V$

$$G\to GL(V)$$

is called a representation of the group $G$. One can classify and study such homomorphisms (upto an equivalence ofcourse) and it’s called representation theory (of finite groups). This “helps” in doing linear algebra when we have a invertible linear map $V\to V$, in my opinion.

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• after doing finite groups and linear algebra it is easy to study representation theory of finite groups any text like Artin or Fulton Harris or Dummit and Foote covers all the introductory theory
• after doing rings and modules the same theory hits different, rephrase everything about representations of a group $G$ with its group algebra $\mathbb{C}[G]$
• after doing commutative algebra, I feel it will give another vibe

Lie groups §

Lie groups are famous. There are different levels to study Lie groups:

• after doing finite groups one may try to study matrix Lie groups, using knowledge of multivariable calculus, it must not be ideal
  • Lie Groups and Lie Algebras - YouTube
  • Borcherds always a good idea Lie groups - YouTube
• after doing topology, smooth manifolds you must already know definitions of topological and Lie groups along with their Lie algebras and exponential map
• after doing algebraic topology you will start understanding what simply connected Lie groups allow that its quotients by discrete subgroups don’t

representation theory of Lie groups §

We use Lie algebras to do representation theory of Lie groups.

representation theory of topological groups §

various groups in topology, geometry and dynamics §

• Homotopy and homology groups
• Riemannian isometry groups
• Symplectic groups

Using algebraic topology to study groups §

We can use the ideas of fundamental groups and covering spaces to say somethings about some infinite groups.

Using differential or other geometries to study groups §

“geometric group theory”, “Fuchsian groups”