Question

let N is a normal cyclic subgroup of a group G,then prove that every subgroup of N is normal in G.

Answer 1

Answer on Question #52825 – Math – Abstract Algebra

Let N be a normal cyclic subgroup of a group G, then prove that every subgroup of N is normal in G.

Solution

Since H is a subgroup of N, if $h$ is in H, then $h = x^k$ for some integer $k$. For any $g$ in G, we have $ghg^{-1} = g(x^k)g^{-1} = (gxg^{-1})^k$. Since N is normal, $gxg^{-1}$ is again in N, say $gxg^{-1} = x^m$, for some integer $m$.

Therefore,

$$g x g^{-1} = (g x g^{-1})^k = (x^m)^k = x^{mk} = x^{km} = (x^k)^m = h^m.$$

Now $h^m$ is in $\langle h \rangle$, which is contained in H (H is closed under multiplication), therefore, $ghg^{-1}$ is in H, that is, $gHg^{-1}$ is contained in H, that is, H is normal.

www.AssignmentExpert.com