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5h
awarded  Disciplined
Sep
9
revised How is $A \implies B$ different from $B \implies A$
edited title
Sep
9
revised If $G$ is a finite group whose $p$-Sylow subgroup $P$ lies in its center, then there is a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$
edited title
Sep
9
comment When can a homomorphism be determined entirely by its generators
@WantTobeAbstract Your map is for all group elements, not just the generators! If it was for just the generators, then you first need a presentation of your group $G$. The relations are the relations of this presentation. For example, If $G$ was the group $\mathbb{Z}_6\times\mathbb{Z}_2=\langle a, b; ab=ba, a^6=1, b^2=1\rangle$ and $n=2$ then you need to verify that $a^2b^2=b^2a^2$, that $a^{12}=1$ and that $b^4=1$. These three relations all clearly hold, so the map $a\mapsto a^2$, $b\mapsto b^2$ is a homomorphism.
Sep
8
comment When can a homomorphism be determined entirely by its generators
@WantTobeAbstract Ah, okay, you just need to check that the homomorphism works for the relators of a given presentation (with your given generators). You might find this answer of mine useful.
Sep
8
comment When can a homomorphism be determined entirely by its generators
I am unsure precisely what you are asking, but I think the restriction you are after is if your map is $\phi: G\rightarrow H$ then you want the generators to generate a subgroup of $H$ which is isomorphic to a homomorphism of $G$. In my first example, $\mathbb{Z}$ contains no elements of finite order but all homomorphic images of $\mathbb{Z}_n$ are finite cyclic. In the third example, $G$ has only itself and the trivial group as a homomorphic image, but $H$ does not contain a copy of $G$.
Sep
8
revised When can a homomorphism be determined entirely by its generators
added 412 characters in body
Sep
8
comment When can a homomorphism be determined entirely by its generators
So does my answer help or not? If not, I'll delete it.
Sep
8
comment When can a homomorphism be determined entirely by its generators
Are you asking "when does every choice of generator produce a homomorphism"? This is different from your states question (but is more interesting!).
Sep
8
answered When can a homomorphism be determined entirely by its generators
Sep
3
revised Calculus: L′ Hopital's Rule
deleted 18 characters in body
Sep
3
reviewed Leave Closed $M=\{a+b\sqrt{2}: a,b \in \mathbb{Q} \}$ and $N=\{c+d\sqrt{3}: c,d \in \mathbb{Q}\}$. $M \cap N \subseteq \mathbb{Q}$.
Sep
3
comment Express a given integer in terms of given, smaller, integers.
Ah, of course, division will give an answer if and only if $m_r=1$ (equivalence because the $a_i$ can be $0$). But the answer is not necessarily optimal.
Sep
3
reviewed Leave Open Question on elements of a set
Sep
3
reviewed Close About the absolute inequality of variables $x,y,z$
Sep
3
reviewed Looks OK How can I do this? $\int\frac{dx}{x^4+1}$
Sep
3
reviewed Looks OK How to evaluate $I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$
Sep
3
reviewed Looks OK How to find the least perfect square bigger than $2b^2$?
Sep
3
reviewed Looks OK tempered distribution and sobolev spaces
Sep
3
reviewed Looks OK Integration using substitution question?