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Comments Off on Download: Bleach Bankai Revolution v2.0 FINALQ:

Multiple Cosets of a group.

Let $G$ be a group with $H,K$ subgroups of $G$.
We say that
(a) $H$ and $K$ are semidirectly subgroups if for every $x\in G$
there exists $y\in H,g\in K$ such that $x=yg$
(b) $H$ and $K$ are left cosets of $H$ respectively of $K$
if there exists $x\in G$ such that $H\circ x=K$ or $K\circ x=H$
My question:
Now I have to check if
(i) $H$ and $K$ are semidirectly subgroups or cosets
(ii) If $H$ and $K$ are semidirectly cosets of $H$ or $K$, then $H$ and $K$ are semidirectly subgroups of $G$
My Attempt:
For $(i)$
I claim that
$H\circ x=\{h_1x\mid h_1\in H\}$ or $K\circ x=\{k_1x\mid k_1\in K\}$
This i have managed to do. But I think i can do that for the other direction, too.
As for $(ii)$ I have no idea how to do that. I am looking forward to any hints or help.

A:

For (i): If $H$ and $K$ are cosets of $H$ and $K$ respectively, then $Hx = \{hx: h \in H\}$ and $Kx = \{kx: k \in K\}$. Thus $H\circ x = Hx$ and $K\circ x = Kx$.
For (ii): Try $x = e$; then $H = \{h: h

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