# Tagged Questions

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### equivalence classes of ∼ are left cosets of H in G - my attempt

Let $H$ be a subgroup of G, and define a relation $∼$ on G by the rules that $x∼y$ mean $x^{-1}y\in H$. Show that $∼$ is an equivalence relation and its equivalence classes are the left cosets ...
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### Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...
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### Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
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### Proper Ideal and Units Proof

Show that an ideal of I R is proper if and only if it does not contain 1 iff and only if it does not contain any units. (1 is the identity element) I'll need to show 3 parts: (1) $\implies$ (2): ...
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### Let H and K be subgroups of the finite group G and supposes $|H|^{2}>|G|$ and $|K|^{2}>|G|$. Prove $H \cap K$ has at least two elements

So I supposed $|H \cap K|>1$ $\Rightarrow |HK||H \cap K|> |HK|$ Which eventually implied that $\Rightarrow |H \cap K|>|G|$ Thus since G is a group, and H and K are subgroups then the ...
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### Finite Abelian groups, G, H, K: $G \times H \cong G\times K$ then $H\cong K$

Let $G,H,$ and $K$ be finite abelian groups then if $G \times H \cong G\times K$ then $H\cong K$. I am trying to use the fundamental theorem for abelian groups to solve this, it is clear intuitively ...
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### Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$ I look up for some ...
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### For any normal subgroup $(aN)^n=(a^n) N$ holds

Prove this theorem Let $G$ be a group and $N$ a normal subgroup of $G$. If $a \in G$ and $n \in Z$, then $(aN)^n = (a^n) N$. I know I should prove this theorem in 3 cases where $n = 0$, $n>0$, ...
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### Prove that the set Aut(G) of all automorphisms of the group G with the operation of taking the composition is a group

Let $G$ be a group. Say what it means for a map $\varphi: G \rightarrow G$ to be an automorphism. Show that the set-theoretic composition $\varphi \psi = \varphi \circ \psi$ of any two ...
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### Prove that any group $G$ of order $p^2$ is abelian, where $p$ is a prime number [duplicate]

Possible Duplicate: Showing non-cyclic group with $p^2$ elements is Abelian "Let $p$ be a prime number. Prove that any group $G$ of order $p^2$ is abelian. You may assume the fact that the ...
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This is the proof, which I mostly understand except for one bit: You have $h_1 \in H_1$ and $h_2 \in H_2$. We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in ... 0answers 156 views ### Easy proof of trivial fusion implies normal p-complement Theorem: Suppose G is a finite group with Sylow p-subgroup P. Then the following are equivalent: The set K of elements of G of order relatively prime to p (the p′-elements) form a subgroup If A and ... 1answer 1k views ### The Product of Subgroups of an Abelian Group Reference: Fraleigh p. 58 Question 5.43 in A First Course in Abstract Algebra Let$G$be an Abelian Group. Suppose$H$and$K$are subgroups of$G$, and$HK = \{ xy: x \in H \text{ and } y \in K ...
In the following problems, let $G$ be an Abelian group. 1) Let $H = \{ x \in G: x=y^{2} \text{ for some } y \in G \}$; that is, let $H$ be the set of all the elements of $G$ which have a square root. ...