Isomorphism

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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; +).

My answer was (A)..

Please help (Crying)
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; +).

My answer was (A)..

Please help (Crying)

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

Tonio
Oct 25th 2010, 10:34 PM
Dreamer78692

Sorry about that...

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; +).

My answer was (A)..

Please help (Crying)

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.