Step 1 Here is an alternative answer, trying to bring structure in the game. We have the short exact sequence: 0→6Z18Z⏟H→Z18Z⏟G→Z6Z⏟GH→0 The image of that complicated element 5+18Z∈G inside GH, so (5+18Z)+H=(5+18Z)+(6Z+18Z)=(5+6Z)∈GH is simply 5+6Z, an element of order six

Step 1 Observe, that H={−6+18Z,18Z,6+18Z}. Hence, |GH|=[G:H]=18:3=6. Now: 5―=5+H={−1+18Z,5+18Z,1118Z} 5―+5―=10+H={4+18Z,10+18Z,−2+18Z} 5―+5―+5―=15+H={9+18Z, 15+18Z,3+18Z} None of these is H. But you can easily calculate 5―+5―+5―+5―+5―+5― Is it H or not.

Step 1 Here is an alternative answer, trying to bring structure in the game. We have the short exact sequence: 0→6Z18Z⏟H→Z18Z⏟G→Z6Z⏟GH→0 The image of that complicated element 5+18Z∈G inside GH, so (5+18Z)+H=(5+18Z)+(6Z+18Z)=(5+6Z)∈GH is simply 5+6Z, an element of order six

Step 1 Observe, that H={−6+18Z,18Z,6+18Z}. Hence, |GH|=[G:H]=18:3=6. Now: 5―=5+H={−1+18Z,5+18Z,1118Z} 5―+5―=10+H={4+18Z,10+18Z,−2+18Z} 5―+5―+5―=15+H={9+18Z, 15+18Z,3+18Z} None of these is H. But you can easily calculate 5―+5―+5―+5―+5―+5― Is it H or not.

Find out the answer to "Find the order 5+H"

College
Algebra
Abstract algebra

kihanja20

Answered question

2022-01-12

Find the order

5 + H

Answer & Explanation

censoratojk

Beginner2022-01-13Added 46 answers

Step 1
Here is an alternative answer, trying to bring structure in the game. We have the short exact sequence:
$0 \to \underset{H}{\underset{⏟}{\frac{6 Z}{18 Z}}} \to \underset{G}{\underset{⏟}{\frac{Z}{18 Z}}} \to \underset{\frac{G}{H}}{\underset{⏟}{\frac{Z}{6 Z}}} \to 0$
The image of that complicated element $5 + 18 Z \in G$ inside $\frac{G}{H}$ , so
$(5 + 18 Z) + H = (5 + 18 Z) + (6 Z + 18 Z) = (5 + 6 Z) \in \frac{G}{H}$
is simply $5 + 6 Z$ , an element of order six