The right way of group theorists thinking This question is invoked by a typical problems on abstract algebra when I tackled.
Question: 

Give an example of a subgroup $H$ of a group $G$ and an element $u$ such that $uHu^{-1}<H$ (Strictly contained but not equal)

My first idea flow to me is trying to find a group satisfy the following group presentations: An infinite group which is generated by $\sigma$ and $\tau$ such that $\sigma\tau\sigma^{-1} = \tau^2$
However, I found difficult in order to find a concrete example but only with group presentations form.
My question is, How group theorists will think when they desire to find an example of a group satisfy a given condition. Are they prefer to find a present example directly or they prefer to think indirectly, i.e. first try a group with presentations, then try to give a concrete example.
My question may be meaningless, but all I want to know is how will they do when solving these similar problems. SINCE I FOUND THAT IT WOULD BE EASIER TO CREATE A GROUP PRESENTATION RATHER THAN FIND A CONCRETE EXAMPLE STRAIGHTLY INSTEAD!
Sorry for my bad English.
 A: I claim that there there is no "one" way, no "correct" way for a group theorist to think. If we all thought the same way then there would be no need to collaborate, and no problems would ever be solved (as either everyone could do them, 'cause we all think the same way, or no one could, for the same reason!). Indeed, geometric group theory is a subject which finds its strength by taking people who think differently (in the sense that they use different tools) and mashing them together.
As for the specific question, I believe that there are two ways of approaching it.


*

*Build a group from scratch, using a presentation.

*Find a group lying around with these properties.


Neither way is "more correct" that the other. Let me give two examples of hard problems which have been solved using one, but not both, of the two methods.
Example 1. In the late 70s/early 1980s, Ol'shanskii proved that there exist "Tarski monster" groups. These are finitely generated, infinite groups all of whose proper subgroups are finite cyclic groups of order a fixed prime $p$. He constructed these groups using presentations and the tools of combinatorial and geometric group theory.
Example 2. M. Kapovich proved constructed non-linear hyperbolic groups (see Section 8 of Kapovich, Michael. "Representations of polygons of finite groups." Geometry & Topology 9 (2005): 1915-1951.). His idea was to consider a hyperbolic group which is a quotient of a uniform lattice in a quaternionic hyperbolic space $\mathbb{HH}^n$, $n \geq 2$, and then apply superrigidity. This is interesting because hyperbolic groups are often constructed using presentations, but despite Kapovich's construction being rather explicit, no presentations are involved.
