# 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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### Solid whose rotational symmetry group corresponds to $\textrm{SO}(2)\times \mathbb{Z}_2$

Sorry to inundate the feed with a question quite similar to my last, but again I've been drawing pictures for quite a while with little success. Does anyone have any idea how to represent the product ...
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### How to write a rigorous proof for bijection?

Let $H$ be a subgroup of a group G. Show that if $aH=bH$ then $Ha^{-1}=Hb^{-1}$. Prove that the map $\alpha$ defined by $\alpha(gH)=Hg^{-1}$ is a bijection. For the first part of the proof: ...
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### Group actions and cardinality of double cosets

I was recently asked this in my abstract algebra class on group actions which seems difficult for me and so need the help on: Let $G$ be a group and $H,K \leq G$ be two subgroups with the ...
### Order of group $G=\langle x,y\,\,|\,\,x^2=y^2=(xy)^3=1\rangle$.
Find the order of group $G=\langle x,y|x^2=y^2=(xy)^3=1\rangle$. I find that $|G|=5$ but this is not possible because $|x|=2$ not divides $|G|=5$. Thanks for your help.