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)

Quote:

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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)

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

Tonio

Sorry about that...

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

A - Alternating Groups

Quote:

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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)

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

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 $\displaystyle D_4$ is abelian but not cyclic).

Quote:

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Originally Posted by Swlabr

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

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 $\displaystyle D_4$ is abelian but not cyclic).

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

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

Tonio

Quote:

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Originally Posted by tonio

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

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

Tonio

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

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

Quote:

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Originally Posted by Swlabr

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

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

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

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

Tonio

Quote:

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Originally Posted by tonio

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

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

Tonio

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Fair point. In which case, (A) is false.

Quote:

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Originally Posted by Swlabr

Fair point. In which case, (A) is false.

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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

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.

Quote:

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Originally Posted by MichaelMath

...I wish they would stop referring to math as "maths" in the UK.

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