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dolphi86 [110]

3 years ago

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Mathematics

1 answer:

Vinil7 [7]3 years ago

8 0

15.38462 is the answer i think

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What is the slope of this line

Rashid [163]

Answer:

Slope = 2/3

Step-by-step explanation:

Slope = Change over y (rise) over change over x (run)

Pick two dot points on the line

Count how many units over they are until you get to the next dot, then count how many units up until you reach the next dot

Remember: Y runs vertically, X runs horizontally

Put the unit number that ran vertically on the top, then put the unit number that runs horizontally on the bottom

If needed simplify

The slope is 2/3

5 0

3 years ago

Prove that H c G is a normal subgroup if and only if every left coset is a right coset, i.e., aH = Ha for all a e G

Kaylis [27]

$\Rightarrow$

Suppose first that $H\subset G$ is a normal subgroup. Then by definition we must have for all $a\in H$ , $xax^{-1} \in H$ for every $x\in G$ . Let $a\in G$ and choose $(ab)\in aH$ ( $b\in H$ ). By hypothesis we have $aba^{-1} =abbb^{-1}a^{-1}=(ab)b(ab)^{-1} \in H$ , i.e. $aba^{-1}=c$ for some $c\in H$ , thus $ab=ca \in Ha$ . So we have $aH\subset Ha$ . You can prove $Ha\subset aH$ in the same way.

$\Leftarrow$

Suppose $aH=Ha$ for all $a\in G$ . Let $h\in H$ , we have to prove $aha^{-1} \in H$ for every $a\in G$ . So, let $a\in G$ . We have that $ha^{-1} =a^{-1}h'$ for some $h'\in H$ (by the hypothesis). hence we have $aha^{-1}=h' \in H$ . Because $a$ was chosen arbitrarily we have the desired .