# Tagged Questions

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### When is a non-trivial homomorphism injective?

I noticed that over the natural numers $(\mathbb{Z},+)$ any group homomorphism $f : \mathbb{Z} \rightarrow \mathbb{Z}$ that is not the trivial one, is automatically injective. Where exactly does ...
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### Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
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### Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
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### Show $D_3\cong S_3$ and $D_n\ncong S_n$ for $n\gt 3$

Show that $D_3\cong S_3$ and $D_3\ncong S_3$ for $n\gt 3$, where $D_3$ denotes the dihedral group and $S_3$ the symmetric group. I define a group isomorphism between $D_3$ and $S_3$. Both group ...
### Finite Subgroups of $GL_2(\mathbb Q)$
I want to prove that the only finite subgroups of $GL_2(\mathbb Q)$ are $C_1, C_2, C_3, C_4, C_6, V_4, D_6, D_8,$ and $D_{12}$. First, we determine all possible finite orders of elements. Now, an ...