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C In parts 1-5 below, $G$ is a group and $H$ is a normal subgroup of $G$. Prove the following (Theorem 5 will play a crucial role)

Theorem 5- Let G be a group and H be a subgroup of G. Then

$(i)$ $Ha= Hb\quad\text{iff}\quad ab^{-1}\in H$

$(ii)$ $Ha = a\quad\text{iff}\quad a\in H$

  1. if $x^2\in H$ for every $x\in G$, then every element of $G/H$ is its own inverse. Conversely, if every element of $G/H$ is its own inverse, then $x^2\in H$ for all $x\in G$.

  2. Suppose that for every $x\in G$, there is an integer $n$, such that $x^n\in H$; then every element of $G/H$ has finite order. Conversely, if every element of $G/H$ has finite order, then for every $x\in G$ there is an integer $n$, such that $x^n\in H$.

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    For $(ii)$, do you mean $Ha=H$?2012-11-15

1 Answers 1

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You must understand how the quotient group $\,G/H\,$ is defined when $\,H\triangleleft G\,$ , though you don't need normality to prove (i)-(ii): this is true for any subgroup.

Now, if $\,x\in G\,$ then, by definition, in the quotient $\,G/H\,$ we have

$(xH)^n:=x^nH\Longrightarrow (xH)^2=x^2H=H\Longleftrightarrow x^2\in H$

and this should get you started for the other questions.