Show that all the cosets of a subgroup H of a group G have t

Show that all the cosets of a subgroup H of a group G have the same number of elements, that is

o(xH) = o(H) = o(Hx) for every x ? G .

Find all the cosets of the subgroup H = {0, 3} of ?3 of the rigid movements of the equilateral

triangle.

Show that all the cosets of a subgroup H of a group G have the same number of elements, that is o(pH) = o(H)-(Ha) for every x E G Find all the cosets of the subgroup H = {03 0A3 of the rigid movements of the equilateral triangle

Solution

Let G be a group and x be any element in G

H is a subgroup of G

Left coset xH = {xh\\h belongs to H}

Right coset Hx = {hx\\h belongs to H}

Only if right and left cosets of H coincide we use the notation H as coset of H

This happens only if H is normal

Hence since o(H) denotes that H is normal with left and right cosets the same

it follows that left and right coset coincide with H and hence

o(xH) = o(Hx) = o(H)

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