Question: "Let H be a subgroup of G and..." - Plainmath

College
Algebra
Abstract algebra

Madeline Lott

Answered question

2022-01-12

Let H be a subgroup of G and

x, y \in G

. Show that

x (H y) = (x H) y

Answer & Explanation

temnimam2

Beginner2022-01-13Added 36 answers

Step 1
This can be done in a few lines:

x (H y) = {x h m i d h \in H y}

= {x (h y) ∣ h \in H}

= {(x h) y ∣ h \in H}

= {h y ∣ h \in x H}

= (x H) y

Pademagk71

Beginner2022-01-14Added 34 answers

Step 1
If

w \in x (H y)

then for some

v \in H y

So

w = x h y

So for some

u \in x H, w = u y

Thus

w \in (x H) y

Therefore

x (H y) \subseteq (x H) y

The inclusion in the other direction can be shown in the same way.

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