Possible Hint: Let H act on G/H by left multiplication. The coset H exist in G/H is fixed by this
action. By considering the possible number of points in a orbit, show that the other elements of
G/H are also left fixed by the action.
Solution
p is the smallest prime that divides n the order of G
H is a subgroup such that [G:H] = p.
Let H act on G/H by left multiplication
Then the coset H exist in G/H is fixed by this.
Consider all possible points in an orbit, for all we find that G/H are also fixed by the action.
As g belongs to G and gH is in G/H, we can see that
g-1Hg is contained in H.
As our selection of g was arbitrary, this is true for all g.
Hence H is normal..
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Possible Hint Let H act on GH by left multiplication. The coset H .pdf
1. Possible Hint: Let H act on G/H by left multiplication. The coset H exist in G/H is fixed by this
action. By considering the possible number of points in a orbit, show that the other elements of
G/H are also left fixed by the action.
Solution
p is the smallest prime that divides n the order of G
H is a subgroup such that [G:H] = p.
Let H act on G/H by left multiplication
Then the coset H exist in G/H is fixed by this.
Consider all possible points in an orbit, for all we find that G/H are also fixed by the action.
As g belongs to G and gH is in G/H, we can see that
g-1Hg is contained in H.
As our selection of g was arbitrary, this is true for all g.
Hence H is normal.
![Possible Hint: Let H act on G/H by left multiplication. The coset H exist in G/H is fixed by this
action. By considering the possible number of points in a orbit, show that the other elements of
G/H are also left fixed by the action.
Solution
p is the smallest prime that divides n the order of G
H is a subgroup such that [G:H] = p.
Let H act on G/H by left multiplication
Then the coset H exist in G/H is fixed by this.
Consider all possible points in an orbit, for all we find that G/H are also fixed by the action.
As g belongs to G and gH is in G/H, we can see that
g-1Hg is contained in H.
As our selection of g was arbitrary, this is true for all g.
Hence H is normal.](https://image.slidesharecdn.com/possiblehintlethactonghbyleftmultiplication-230326023439-23912896/85/Possible-Hint-Let-H-act-on-GH-by-left-multiplication-The-coset-H-pdf-1-320.jpg)
