# Some questions about mathematics [closed]

This question is a soft one. Well, So far I have noticed stuff that is nice in math, particularly in algebra, topology and analysis. For instance, in algebra, there is theorem that says that we can think of groups just as some set of permutations. So, in other words, can we say all groups are just permutations? Also in topology we classify surfaces. In fact, we have that every compact connected surface is either a sphere, an n-torus of $n-$ projective planes. Is there any similarity in measure theory? It seems like math is just like comparing things. Is this true? Also, one last question, What is the big picture that everyone talks about?

thanks

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## closed as too broad by Zev Chonoles, Johannes Kloos, Matthew Pressland, martini, SiméonOct 10 '13 at 8:22

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs. If this question can be reworded to fit the rules in the help center, please edit the question.

This is an unreasonably broad question, and it is difficult to understand what you are asking. I've voted to close. –  Zev Chonoles Oct 10 '13 at 7:58

• All groups are permutation groups, in the following sense: Any group $G$ acts on itself by left multiplication. In other words, for any $g\in G$, the map $h\mapsto gh$ is a permutation of $S=|G|$, the underlying set of $G$. This gives an injective homomorphism $G\to Sym(S)$. For this reason, we can think of any group as a permutation group (though this is not necessarily a useful way to think about all groups).
• There are many classification theorems in measure theory (and in all areas of mathematics). One simple one is the correspondence between Borel measures on $\mathbb{R}$ and increasing, right-continuous functions $\mathbb{R}\to\mathbb{R}$. Another is the uniqueness of Haar measure.