In understanding unitary group, i get confused because there are several definition of unitary group, first, in here: [Sven Grützmacher](http://www.mathi.uni-heidelberg.de/~lee/SvenSS16.pdf)

Let A matrix and define $A^{\*}=\bar{A}^{T}$, Then we can define the unitary group,

$U(n)=\{M \in M\_n \mathbb{(C)} | M^\*M=I\}$

Then, 

$U(p,q)=\{M \in M\_n \mathbb{(C)} | M^\*I\_{p,q}M=I\_{p,q} \}$

is the indefinite unitary group of signature $(p,q)$, where $p+q=n$.

But, in here: [Order of finite unitary group](https://mathoverflow.net/questions/34387/order-of-finite-unitary-group)

$U\_n(q)= \{ M \in GL\_n(F\_{q^{2}}) | MM^\*=I\_n \}$

Also, from the above link and the book "The Subgroup Structure of The Finite Classical Groups", known the order of finite unitary group to be:

$q^{(n^2-n)/2} \prod \_{k=1} ^{n} (q^k - (-1)^k)$.

So, if we take example of unitary group $U(3,5)$, according to the first link, the member of the group is $8\times8$ matrices? and according to the second link, the member of the group is $3\times3$ matrices over Finite Field in a of size 5^2? and when i used GAP, the order of U(3,5) is 126000, but with sage, is 228000. And, by manual counting, with $n=3$ and $q=5$, got 228000.

Would you give me clear definition of unitary group, definite and indefinite unitary group?

And, in paper "Unitals, projective planes and other combinatorial structures constructed from the unitary groups $U(3,q), q=3,4,5,7$", for q=5, which one is used? Because, in the paper, $U(3,5)$ have order of 126000, how they get it? only using GAP? how about the manual?

note that: i have not studied lie group yet.