Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Retractions and Isomorphisms of Fundamental Groups

Suppose there is a retraction from $$S^1 \times D^2 \to S^1 \times S^1.$$ Does that then induce an isomorphism $$\pi_1(S^1) \times \pi_1(D^2) \cong \pi_1(S^1) \times \pi_1(S^1)?$$ Which is obviously ...
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Is the group of units in $\mathcal{O}_{\mathbb{Q}(\sqrt{2})}$ finite and cyclic?

Let $K=\mathbb{Q}(\sqrt{2})$ and let $\mathcal{O}_K$ be its ring of integers. Consider the group of units in $\mathcal{O}_K$. Is it finite? Is it cyclic? My thought: For any $\alpha = a+b\sqrt{2}$, ...
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Isomorphism between semidirect products

Alperin and Bell, Groups and Representations, section 2, Proposition 11, p. 23, states: "Let $H$ be a cyclic group and let $N$ be an arbitrary group. If $\varphi$ and $\psi$ are monomorphisms from $H$ ...
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Does the converse of Lagrange's theorem hold for any finite Dedekind group?

I'm currently reading a book on algebra and the following argument is used to prove that the converse of Lagrange's theorem (if $d$ divides $|G|$ there exists a subgroup of order $d$) holds for any ...
I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...