I was getting ready to learn about order of an element, cosets, and langrange's theorem in group theory. Consequently this involves multiplying two elements of a group. After seeing several examples more abstract than multiplication of matrices, fields, and etc like with symmetries, I'm guessing multiplication in general with groups is just composition, but I've been unable to confirm that. From various other things I've already developed a mindset of multiplication (in ANY sense) and composition being the same, but I wanted to check this is also true in group theory, or concretely if a group $G$ with operation $*$ and elements $a,b,c$ that multiplication $ab$ can be defined as:
$ab=a*b$ which follows intuitively from: $a*b*c=(ab)*c$ or $a*b*=(ab)*$
...but this does seem odd we'd introduce the new notion of multiplication here if we already have the group operator.
Forgive me if this is a duplicate and I didn't look enough.
EDIT: clarification, I don't know weather or not there's a general way to multiply two elements of a group (to add context, like you do in finding a coset or when raising a group element to the nth power to find it's order). If there is a general meaning of multiplication, does it basically just work like function composition, or equivelently whatever the group operator is?