Question

let a be a fixed element of a group G. show that H={ax=xa , where x is the element of G} is a subgroup of G.

Answer 1

Let $a$ be a fixed element of a group $G$. Show that $H = \{x \in G : xa = ax\}$ is a subgroup of $G$.

**Solution.**

Check that $\forall x, y \in H: x^{-1} \in H, x \cdot y \in H$:

Let $x \in H$.

$$\begin{array}{l} xa = ax \Rightarrow x^{-1}(xa) = x^{-1}(ax) \Rightarrow a = x^{-1}ax \Rightarrow ax^{-1} = (x^{-1}ax)x^{-1} \Rightarrow \\ \Rightarrow ax^{-1} = x^{-1}a \Rightarrow x^{-1} \in H; \end{array}$$

Let $x, y \in H$.

$$ya = ay \Rightarrow xya = x(ay);$$ $$xa = ax \Rightarrow x(ay) = (ax)y;$$

Hence:

$$xya = axy \Rightarrow xy \in H.$$

Thus,

1) $\forall x \in H: x^{-1} \in H$;

2) $\forall x, y \in H: xy \in H$;

Besides:

$\forall x \in H: x \cdot x^{-1} \in H \Rightarrow e \in H$ ($e$ - neutral element);

$\forall x, y, z \in H: x, y, z \in G \Rightarrow x \cdot (y \cdot z) = (x \cdot y) \cdot z$ (associativity);

So, $H$ is a subgroup of $G$.