Question:
If $G$ is a group and $H<G$ and $|G/H|=2$ prove that for each $a\in H$, $aH=Ha$
Can someone please provide some imagination of $G/H$? ( And also provide at least a hint on the question? )
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Not much imagination on $G/H$ : it's a group (since you are asked to prove that $H\triangleleft G$) and it has two elements. One element (for a quotient group it's always the coset of $1$) is the unity, the other one is the other one. – Mar 03 '17 at 02:42
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Do you mean "for each $a\in G$" rather than $a\in H$? If $a\in H$ then we have $aH=H=Ha$ as $H$ is a group. – JonCC Mar 03 '17 at 02:50
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@JonCC no:) for each $a \in H$ is correct. – Maryam Seraj Mar 03 '17 at 02:57
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You have $|G/N|=2$ then you only have 2 distinct cosets.
Lets check right cosets one corresponds to $N$ and you know cosets are disjoint so the other one must be $Na $ for some $a\notin N $ and $Na=G-N$.
Now check the left cosets, again you have the one corresponding to $N $ and you know $a $ is not in $N $ so $aN$must be distinct to $N $ now you have that $aN=G-N $ and then $Na=aN $ and $N=N $ and the claim is proved.
Note: $N$ as coset denotes the class of $Ne$ where $e $ is the identity element of $G $
Jonathaniui
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