Resolved: "Why must a be in one of these..."

sunshine022uv

Answered question

2022-01-13

Why must a be in one of these two cosets:

H, H a ?

Thomas Lynn

Beginner2022-01-14Added 28 answers

Step 1
Remember that the cosets are exactly the equivalence classes of the equivalence relation

a \sim b \Leftrightarrow b^{- 1} a \in H

.
We know that if we define an equivalence relation on a set X, the disjoint union of the equivalence classes is exactly X.
So

G = H \cup a H

.
Since

a^{2} \in G

, it must be in one of them.

Esta Hurtado

Beginner2022-01-15Added 39 answers

Step 1
H has index

2 = \frac{600}{300}

in G, hence it is normal in G and

\frac{G}{H}

is isomorphic to the group of order 2, hence for every

a \in G, a^{2} H = H,

a^{2} \in H

alenahelenash

Expert2022-01-24Added 556 answers

HINT: Note that H and Ha cannot intersect, otherwise there would be a \(h,hin

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