Looking for some insight or explanation in Normal group and similar matrix I need someone can give me some insight in Normal group and Matrix Similarity.
Normal subgroup
$g \in G \text{ and } N \lt G$ 
$gNg^{-1} = N$
Matrix Similarity
$B = PAP^{-1}$
It seems to me Normal subgroup and Matrix Similarity have "similar" structure. 
I mean $PAP^{-1}$ and $gNg^{-1}$
Can anyone explain to me, is there any direct or indirect relationship between Normal group and Matrix Similarity?
 A: Yes, absolutely!
Only because of the wording, I would say "indirect relationship." By this I mean they're both definitions that rely on conjugation, and that's the main idea here. But a normal subgroup is a relationship between a group and its subgroup, while matrix similarity is a relationship between two matrices.
We can make a "better" analogy by focusing only on the specific relationship between two individual elements, and not bringing subgroups into play.
So, in a generic group $G$, we say that two elements $g, h \in G$ are in the same conjugacy class if there exists some $x \in G$ with $h = xgx^{-1}$.
When our group happens to be a matrix group (e.g., the general linear group of invertible $n \times n$ matrices with entries in some field), saying that two invertible matrices $A$ and $B$ are similar is just saying that they're in the same conjugacy class, in the group-theoretic sense: The two notions completely coincide. 

Historically, matrix algebra came before abstract groups, so it's almost certain that the "similarity" idea and language came first, and only later did we develop a more general setting for conjugacy.
Also, it's worth noting that matrices $A$ and $B$ may be similar yet not invertible. We would still be using conjugation by an invertible matrix, but it wouldn't be accurate to say that everything is taking place in a group anymore.
