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Mathematics

1 answer:

adoni [48]2 years ago

7 0

Answer:

- Early in the race,_Alberto_ runs faster.

- Alberto maintains a speed of _20_ km/h during the first hour of the race.

- meanwhile, Bianca’s running speed _Is exponential_

-The first person to cross the finish line is_Bianca_

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What is the value of 6 times 7 - 3^2 times 9 + 4^3<br> show work..THX

Kitty [74]

Step-by-step explanation:

6 times7-3^2 times 9+4^3=

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(6(7-9)) *(9+64) =

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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 .