In the froup Z_12, find |a|, |b|, and |a+b|a=5, b=4

Prove that in any group, an element and its inverse have the same order.

Is there any relation between ${\displaystyle {\text{Sym}}_n}$ and ${\displaystyle {\text{Sym}}_k}$ where ${\displaystyle k<n}$? Is ${\displaystyle {\text{Sym}}_k\subset {\text{Sym}}_n}$? Since ${\displaystyle k!\mid n!\mathrm{\forall }k<n}$ they satisfy Lagrange's theorem, but of course that doesnt guarantee the existence of a subgroup

Let (Z,+) be a group of integers and (E,+) be a group of even integers. Find and prove if there exist an isomorphism between them.

Simple question about finitely generated algebras
I have two finitely generated algebras A and B over a field K such that ${\displaystyle B\subseteq A}$. Is it true that ${\displaystyle A=B[a_1,\cdots ,a_n]}$ for some ${\displaystyle a_1,\cdots ,a_n\in A}$?

Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spaces
Let U be a unitary transformation from Hilbert space H to Hilbert space K.
How do you call a *-homomorphism f from B(H) to B(K), defined by ${\displaystyle f\left(a\right)=Ua{U}^{-1}}$?
I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech.

Residue of x in polynomial ring ${\displaystyle \mathbb{Z}\frac{x}{f}}$
When we say that α is the residue of x in ${\displaystyle \mathbb{Z}\frac{x}{f}}$, and ${\displaystyle f=x^4+x^3+x^2+x}$, wouldn't ${\displaystyle \alpha }$ just be x? Because if we divide with the remainder, we would get ${\displaystyle x=0f+x}$?

Show that the Lorentz boosts,
$\left(\begin{array}{cc}\gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{array}\right)$
form a one-parameter Lie group.