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The title kind of says it all: I am suppose to determine whether $(1 2) H = (1 3 4)H$ without computing cosets. Where $G = S_4$ and $H$ is an under group defined by: $H = \{e,(1 2 3 4), (1 3)(2 4), (1 4 3 2),(1 3),(1 4)(2 3),(2 4),(1 2)(3 4)\}$

I think the point is that I have to look at $(1 2 3)H$ which I calculated in the previous exercise, where I got: $(1 2 3)H = \{(1 2 3), (1 3 2 4), (1 4 2), (3 4), (1 2), (1 3 4), (1 4 2 3), (2 4 3)\}$

I see that this contains $(1 2)$ and $(1 3 4)$, but I am still not quite sure what this means, and if it tells me anything about $(1 2)H = (1 3 4)H$.

InsideOut
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some_name
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1 Answers1

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In general, $g_1H=g_2H$ (for any group $G$, $g_1,g_2\in G$ and $H$ a subgroup of $G$) if and only if $g_1\in g_2H$.

Ben Sheller
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