I know if G is a group, H is a subgroup of G, and g is an element of G

gH= {gh: h an element of H} is a left coset of H in G

But just cant apply to the Questions Help me !
Thanks

1) consider the element X=(1423) ∈ S₄

ⅰ) Write down the elements of <X>
ⅱ)Write down the elements of the left coset<12><X>
ⅲ)Write down the elements of the left coset<123><X>
ⅳ)Determine the index [S₄: (X)]

2. Originally Posted by Hyungmin
I know if G is a group, H is a subgroup of G, and g is an element of G

gH= {gh: h an element of H} is a left coset of H in G

But just cant apply to the Questions Help me !
Thanks

1) consider the element X=(1423) ∈ S₄

ⅰ) Write down the elements of <X>
ⅱ)Write down the elements of the left coset<12><X>
ⅲ)Write down the elements of the left coset<123><X>
ⅳ)Determine the index [S₄: (X)]
Write down the elements? I'm sure you can do that. Similarly, the next two are just computation. The last can be seen as a result of finding all the cosets or employing Lagrange's theorem considering that $\left|\left\langle (1423)\right \rangle\right|=4$ we have that $\left[S_4:\left\langle (1423)\right\rangle\right]=\frac{4!}{4}$