Isomorphism

• Oct 24th 2010, 11:48 PM
Dreamer78692
Isomorphism
Which of the following statements is true? (V is the Klein-4 group)

"=" represents isomorphic to.

(A) V = R4
(B) A3=R3
(C) A4=R12
(D) V =Z/4Z
(E) (Z; +) = (Q; +).

• Oct 25th 2010, 02:33 AM
tonio
Quote:

Originally Posted by Dreamer78692
Which of the following statements is true? (V is the Klein-4 group)

"=" represents isomorphic to.

(A) V = R4
(B) A3=R3
(C) A4=R12
(D) V =Z/4Z
(E) (Z; +) = (Q; +).

What is R4, R3, R12, A3, A4...??

Tonio
• Oct 25th 2010, 10:34 PM
Dreamer78692

R - stands for the Rotation groups
e.g. R3 = {e, r, r^2}

A - Alternating Groups
• Oct 26th 2010, 12:57 AM
Swlabr
Quote:

Originally Posted by Dreamer78692
Which of the following statements is true? (V is the Klein-4 group)

"=" represents isomorphic to.

(A) V = R4
(B) A3=R3
(C) A4=R12
(D) V =Z/4Z
(E) (Z; +) = (Q; +).

I presume by `rotation group' you mean $D_{n}$, the group of symmetries of a regular n-gon (Rn= $D_n$).

You are correct. What were your thinkings in the other questions? (Order arguments and pointing out cyclic groups when you see them will work). Can you prove your result? (prove that $D_4$ is abelian but not cyclic).
• Oct 26th 2010, 04:02 AM
tonio
Quote:

Originally Posted by Swlabr
I presume by `rotation group' you mean $D_{n}$, the group of symmetries of a regular n-gon (Rn= $D_n$).

You are correct. What were your thinkings in the other questions? (Order arguments and pointing out cyclic groups when you see them will work). Can you prove your result? (prove that $D_4$ is abelian but not cyclic).

Very weird notation, but according to it, and if I didn't misunderstood, also (B) is correct since

both $A_3\,,\,R_3$ are cyclic groups of order 3...

Tonio
• Oct 26th 2010, 04:34 AM
Swlabr
Quote:

Originally Posted by tonio
Very weird notation, but according to it, and if I didn't misunderstood, also (B) is correct since

both $A_3\,,\,R_3$ are cyclic groups of order 3...

Tonio

Notation notation notation! However, I do not think you are correct in any notation. I write $D_n$ for the set of symmetries of a regular n-gon, which is also often written as $D_{2n}$. It is always of even order, and so cannot be cyclic of order 3...

It is true, however, that $D_3 \cong S_3$.
• Oct 26th 2010, 06:06 AM
tonio
Quote:

Originally Posted by Swlabr
Notation notation notation! However, I do not think you are correct in any notation. I write $D_n$ for the set of symmetries of a regular n-gon, which is also often written as $D_{2n}$. It is always of even order, and so cannot be cyclic of order 3...

It is true, however, that $D_3 \cong S_3$.

The OP himself, though, wrote that $R3 = \{e,r,r^2\}=$ a cyclic group of order 3. Perhaps for him "rotation group" is

ONLY the actual rotations (spinnings) , no the whole dihedral group.

Tonio
• Oct 26th 2010, 06:11 AM
Swlabr
Quote:

Originally Posted by tonio
The OP himself, though, wrote that $R3 = \{e,r,r^2\}=$ a cyclic group of order 3. Perhaps for him "rotation group" is

ONLY the actual rotations (spinnings) , no the whole dihedral group.

Tonio

Fair point. In which case, (A) is false.
• Oct 26th 2010, 06:56 AM
tonio
Quote:

Originally Posted by Swlabr
Fair point. In which case, (A) is false.

Indeed, which I think was the case from the beginning, since then one is a cyclic group whereas the other isn't...who knows what the truth is here in Weirdnotationland.

Tonio
• Oct 26th 2010, 08:15 AM
MichaelMath
Sorry to go off topic for a moment, but I dream of a day when all mathematical notation and terminology is standardized internationally. Also, I wish they would stop referring to math as "maths" in the UK.
• Oct 27th 2010, 12:08 AM
Swlabr
Quote:

Originally Posted by MichaelMath
...I wish they would stop referring to math as "maths" in the UK.

That's okay, in the UK we wish they would stop referring to maths as `math' in the US.