In group theory, the action of a group is helpful to determine structure of group. As an example,

If $G$ is of order $p^aq^b$ then $G$ is solvable. [Action considered on vector space.]

If $G$ is a $p$-group then its center is non-trivial. [Action is on non-identity elements by conjugation.]

If $|G|=2m$, $m$ odd, then $G$ contains normal subgroup of order $m$ [Left action of $G$ on itself.]

Beside this, what are interesting theorems which tell structure of group from action of it on some objects (other than vector space and above mentioned type)?

In fact, action of group on vector space has lot of applications to find structure of group. This is in representation theory of finite groups.

In above question, I want to see examples from different (category of) objects for action of group, to determine its structure.

I will be happy to see if something is derived about group by its action on some topological space, some graph, or on complex domain etc.

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    $\begingroup$ I like info obtainable from geometric objects, e.g., the symmetry group of the cube is seen to be $\cong S_4$ - because the defining action on the cube induces an action on the four spacial diagonals $\endgroup$ – Hagen von Eitzen Sep 24 '18 at 6:23
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    $\begingroup$ An example to your last question: a group is free if and only if it admits a free action on some tree. See for example theorem 4.2.1 in these notes: mathematik.uni-regensburg.de/loeh/ggt_book/ggt_book_draft.pdf $\endgroup$ – Mathematician 42 Sep 24 '18 at 7:04
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    $\begingroup$ In a similar vein: I would recommend the brilliant book of Jean-Pierre Serre: Trees. The book is an English translation of "Arbres, Amalgames, SL(2)", published in 1977 by J-P.Serre, and written with the collaboration of H.Bass. The first chapter describes the "arboreal dictionary" between graphs of groups and group actions on trees. The second chapter gives applications to the Bruhat-Tits tree of SL(2) over a local field. $\endgroup$ – Nicky Hekster Sep 24 '18 at 7:43
  • $\begingroup$ you may want to look at vertex operator algebras. $\endgroup$ – Alexander Gruber Oct 11 '18 at 5:54

(1) If a group acting on a finite set is sharply-3/2 transitive (i.e., transitive with non-trivial point stabilizers but only the identity stabilizes two points), then it has a nilpotent regular normal subgroup and the point stabilizer has many restrictions on its structure as well. (These are called Frobenius groups.)

(2) Sharply 2-transitive groups on finite sets are either subgroups of the semi-linear transformations on the elements of a finite field, or on a list of seven exceptions, of degrees (i.e., size of the set being acted upon) $5^2$, $7^2$, $11^2$, $23^2$, $29^2$, or $59^2$. (There are two exceptions of degree $11^2$, one solvable, one not.) All of the solvable exceptions contain a copy of $SL(2,3)$ in the point-stabilizer, and all of the rest contain a copy of $SL(2,5)$ in the point stabilizer. Also, each finite near-field that is not a Dickson near-field corresponds to one of these seven exceptions.

(3) Similar results hold for sharply-5/2 transitive, sharply 3-transitive, etc., groups acting on finite sets. In particular, sharply 3-transitive groups on finite sets are either the group of fractional linear transformations on a projective line for a finite field, or (for fields of size $q^n$ with $q$ odd and $n$ even) the group of semilinear fractional transformations with identity automorphism and square determinant or automorphism of order 2 and non-square determinant, acting on the same projective line. The smallest of the latter (acting on the projective line with 10 points) is the point stabilizer in the Mathieu group $M_{11}$.

(4) Sharply n-transitive groups for $n\ge 4$ must be $S_n$, $S_{n+1}$, $A_{n+2}$ or (for $n=4$) $M_{11}$ or (for $n=5$) $M_{12}$. This result does not depend on the classification the way that a somewhat similar statement with the word "sharply" removed does. In fact, if memory serves, this is a 19th century result due to C Jordan.

(5) In analyzing the structure of finite groups $G$, one often uses theorems that conclude that certain normal subgroups $H$ (e.g., the Fitting subgroup of a solvable group) are self-centralizing. This means that the action of $G/H$ on $H$ is faithful, which can then often be used to obtain more information in specific cases.

(6) There is a good deal of work on simple groups characterizing them in terms of their actions on flags in incidence geometries with various properties. This started with Lie and Lie-type groups, and various analogous geometries have been introduced to characterize sporadic groups.


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