Need help understanding the meaning of these notations

Question

I have these notations in an exercise and I can't understand them, the exercise is in French and I tried to translate it to English.

("e" is the neutral element, "*" is a law of composition) (?)

1) Let (G,*) a group such as:

∀x ∈ G, x²= e

-Is x² <=> x*x ?

2) Let (E,*) a group such as:

∀x ∈ E, x*² = e

-What does x*² mean?

Thank you.

Original material(Ex 1 & 2): http://mp.cpgedupuydelome.fr/pdf/Structures%20alg%C3%A9briques%20-%20Groupes.pdf

Answer 1

Look very carefully at the original wording:

Soit $(G, *)$ un groupe tel que...

So that $G$ is a group. However, in the second exercise:

Soit $*$ une loi de composition interne sur un ensemble $E$ associative et possédant un neutre $e$. On suppose que...

The key word here is "ensemble", which means "set". That is, $E$ is not defined as a group, let alone an abelian group. You must prove that $E$ is a group on $*$.

In the group $G$, the operator is bundled as part of the group, and $x^2$ is unambiguous ($x^2 = x * x$). However, for the case of the set $E$, the term $x^2$ is totally ambiguous. You are told that you have an associative operator $*$ on the set $E$, not that $E$ is a group with the operator $*$. Therefore we require the operator $*$ in the power to distinguish, "You are performing the binary operation $*$ upon the element $x$ two times."

EDIT: You may be interested in this French glossary of math terms if you wish to do this translation. This is an elementary introduction to abstract algebra. Unless you have a reason to prefer this specific set of exercises, you may find it easier to learn from a text written in a language you're fluent with. This introductory material is standard for any course in abstract algebra.

Answer 2

You seem to have difficulties with groups. A group is a set $G$ together with an operation $*$, that combines two elements in $G$ to form a new element in $G$. The group is mostly written like this: $(G,*)$. A group must satisfy certain laws:

1. Closure: When you combine two elements in $G$ with the operation $*$, the third element also has to be part of $G$,
2. Assosiativity: Take 3 elements in $G$: $(a,b,c)$, then the following must apply: $(a*b)*c=a*(b*c)$
3. Neutral element: There exists an element $e$ in $G$, so that for all $x$ in $G$, $x*e = x$
4. Inverse: For all elements $x$ in $G$, there exists an element $x'$ in $G$ so that $x*x'=e$ ($e$ being the neutral element)

An example of a group would be $(\mathbb Z,+)$, the neutral element being $0$, and the inverse beingthe negative value of a given number in $\mathbb Z$. Note that the operation $*$ is not restricted to $+,*$, as any operation that satisfies the group definition works.

Referring to your question, it should now be clear as to what $e$ and $*$ refer, $x^2$ is indeed $x*x$. As for $x^{*x}$, I can not help you, as I do not see what it is supposed to mean. If you have problems with logical quantifiers, refer to https://en.wikipedia.org/wiki/Quantifier_(logic).