Prove that if "a" is the only elemnt of order 2 in a group, then "a" lies in the center of the group.

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

Show that an intersection of normal subgroups of a group G is again a normal subgroup of G

Order of field extension
Im

How can I show these polynomials are not co'?
${\displaystyle x,x-1}$ and ${\displaystyle x+1}$ in the ring ${\displaystyle {\mathbb{Z}}_6\left[x\right]}$

How to find the value of U(n) on abstract algebra
Example U(8) = {1,3,5,7}

Suppose G is a group and H is a normal subgroup of G. Prove or disprove ass appropirate. If G is cyclic, then ${\displaystyle \frac{G}{H}}$ is cyclic.
Definition: A subgroup H of a group is said to be a normal subgroup of G it for all ${\displaystyle a\in G}$, aH = Ha
Definition: Suppose G is group, and H a normal subgruop og G. THe froup consisting of the set ${\displaystyle \frac{G}{H}}$ with operation defined by (aH)(bH)-(ab)H is called the quotient of G by H.

Is k[[x]] ever a finitely generated k[x](x) module?
For k a field, the localization ${\displaystyle k{\left[x\right]}_{\left(x\right)}}$ naturally includes into k[[x]]. I can prove that if ${\displaystyle k=\mathbb{C},}$, then this inclusion is not surjective, and k[[x]] is not even finitely generated over ${\displaystyle k{\left[x\right]}_{\left(x\right)}}$, because ${\displaystyle \mathbb{C}{\left[x\right]}_{\left(x\right)}}$ corresponds to rational functions holomorphic at 0, and ${\displaystyle \mathbb{C}\left[\left[x\right]\right]}$ corresponds to germs of all functions holomorphic at 0, and so with some complex analysis you can see the finite generation is impossible. But over an arbitrary field, is this claim still true?