Statement : Every group of order 6 is cyclic - Proof that the statement is false
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Proof
Let G be a group of order 6. By Lagrange's Theorem, G has subgroups of order 1,2,3 and 6.
The subgroups of orders 2 and 3 have prime orders and are cyclic therefor. The subgroup contains an element g of order 2 and the subgroup contains an element h of order 3.
So therefore G is cyclic.
Is this correct or is there also proof of group G not being cyclic
abstract-algebra group-theory cyclic-groups
$endgroup$
add a comment |
$begingroup$
Proof
Let G be a group of order 6. By Lagrange's Theorem, G has subgroups of order 1,2,3 and 6.
The subgroups of orders 2 and 3 have prime orders and are cyclic therefor. The subgroup contains an element g of order 2 and the subgroup contains an element h of order 3.
So therefore G is cyclic.
Is this correct or is there also proof of group G not being cyclic
abstract-algebra group-theory cyclic-groups
$endgroup$
$begingroup$
Not true, try $D_3$ the dihedral group
$endgroup$
– user25959
Dec 3 '18 at 1:47
$begingroup$
"he subgroup contains an element g of order 2 and the subgroup contains an element h of order 3." but that doesn't mean the order of $hg$ is $6$ or that any element has an order of $6$.
$endgroup$
– fleablood
Dec 3 '18 at 2:03
add a comment |
$begingroup$
Proof
Let G be a group of order 6. By Lagrange's Theorem, G has subgroups of order 1,2,3 and 6.
The subgroups of orders 2 and 3 have prime orders and are cyclic therefor. The subgroup contains an element g of order 2 and the subgroup contains an element h of order 3.
So therefore G is cyclic.
Is this correct or is there also proof of group G not being cyclic
abstract-algebra group-theory cyclic-groups
$endgroup$
Proof
Let G be a group of order 6. By Lagrange's Theorem, G has subgroups of order 1,2,3 and 6.
The subgroups of orders 2 and 3 have prime orders and are cyclic therefor. The subgroup contains an element g of order 2 and the subgroup contains an element h of order 3.
So therefore G is cyclic.
Is this correct or is there also proof of group G not being cyclic
abstract-algebra group-theory cyclic-groups
abstract-algebra group-theory cyclic-groups
edited Dec 3 '18 at 4:26
Kemono Chen
2,9181739
2,9181739
asked Dec 3 '18 at 1:43
winterdeurwinterdeur
62
62
$begingroup$
Not true, try $D_3$ the dihedral group
$endgroup$
– user25959
Dec 3 '18 at 1:47
$begingroup$
"he subgroup contains an element g of order 2 and the subgroup contains an element h of order 3." but that doesn't mean the order of $hg$ is $6$ or that any element has an order of $6$.
$endgroup$
– fleablood
Dec 3 '18 at 2:03
add a comment |
$begingroup$
Not true, try $D_3$ the dihedral group
$endgroup$
– user25959
Dec 3 '18 at 1:47
$begingroup$
"he subgroup contains an element g of order 2 and the subgroup contains an element h of order 3." but that doesn't mean the order of $hg$ is $6$ or that any element has an order of $6$.
$endgroup$
– fleablood
Dec 3 '18 at 2:03
$begingroup$
Not true, try $D_3$ the dihedral group
$endgroup$
– user25959
Dec 3 '18 at 1:47
$begingroup$
Not true, try $D_3$ the dihedral group
$endgroup$
– user25959
Dec 3 '18 at 1:47
$begingroup$
"he subgroup contains an element g of order 2 and the subgroup contains an element h of order 3." but that doesn't mean the order of $hg$ is $6$ or that any element has an order of $6$.
$endgroup$
– fleablood
Dec 3 '18 at 2:03
$begingroup$
"he subgroup contains an element g of order 2 and the subgroup contains an element h of order 3." but that doesn't mean the order of $hg$ is $6$ or that any element has an order of $6$.
$endgroup$
– fleablood
Dec 3 '18 at 2:03
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
To answer your question, it is not correct. Several mistakes here.
Lagrange does not say that such subgroups exist. You are incorrectly applying a false converse. Any subgroup of a group of order $6$ must have order $1, 2, 3$, or $6$. This does not say that such subgroups must exist.
Also, a group $G$ of order $6$ having subgroups that are cyclic does NOT mean that the group $G$ itself is cyclic. In fact, all groups have cyclic subgroups by definition (and usually plenty of them).
If you think the statement is false, then your job is to find a group of order $6$ that is not cyclic. If you think it is true then you must prove that all such groups are cyclic. How many groups of order $6$ do you know?
Edit: never mind, answer given away in the comments.
$endgroup$
$begingroup$
Thank you for your comment - I will try to continue as I was clearly wrongly stuck in my own Lagrange theorem
$endgroup$
– winterdeur
Dec 3 '18 at 1:49
$begingroup$
I guess with order 2 I can proof that group G itself is cyclic.
$endgroup$
– winterdeur
Dec 3 '18 at 1:50
$begingroup$
To see a small example to illustrate your mistakes more simply, notice that all proper subgroups of $G=mathbb{Z}_2 times mathbb{Z}_2$ are cyclic, yet $G$ is not. (Your mistakes are also common, so don't worry too much now. You are right to try and fix them now.)
$endgroup$
– Randall
Dec 3 '18 at 1:51
$begingroup$
"I guess with order 2 I can proof that group G itself is cyclic. " Which gets you nowhere. The subgroup of order 2 is cyclic but that doesn't mean the group of which it is a subgroup is cyclic.
$endgroup$
– fleablood
Dec 3 '18 at 1:57
add a comment |
Your Answer
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1 Answer
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$begingroup$
To answer your question, it is not correct. Several mistakes here.
Lagrange does not say that such subgroups exist. You are incorrectly applying a false converse. Any subgroup of a group of order $6$ must have order $1, 2, 3$, or $6$. This does not say that such subgroups must exist.
Also, a group $G$ of order $6$ having subgroups that are cyclic does NOT mean that the group $G$ itself is cyclic. In fact, all groups have cyclic subgroups by definition (and usually plenty of them).
If you think the statement is false, then your job is to find a group of order $6$ that is not cyclic. If you think it is true then you must prove that all such groups are cyclic. How many groups of order $6$ do you know?
Edit: never mind, answer given away in the comments.
$endgroup$
$begingroup$
Thank you for your comment - I will try to continue as I was clearly wrongly stuck in my own Lagrange theorem
$endgroup$
– winterdeur
Dec 3 '18 at 1:49
$begingroup$
I guess with order 2 I can proof that group G itself is cyclic.
$endgroup$
– winterdeur
Dec 3 '18 at 1:50
$begingroup$
To see a small example to illustrate your mistakes more simply, notice that all proper subgroups of $G=mathbb{Z}_2 times mathbb{Z}_2$ are cyclic, yet $G$ is not. (Your mistakes are also common, so don't worry too much now. You are right to try and fix them now.)
$endgroup$
– Randall
Dec 3 '18 at 1:51
$begingroup$
"I guess with order 2 I can proof that group G itself is cyclic. " Which gets you nowhere. The subgroup of order 2 is cyclic but that doesn't mean the group of which it is a subgroup is cyclic.
$endgroup$
– fleablood
Dec 3 '18 at 1:57
add a comment |
$begingroup$
To answer your question, it is not correct. Several mistakes here.
Lagrange does not say that such subgroups exist. You are incorrectly applying a false converse. Any subgroup of a group of order $6$ must have order $1, 2, 3$, or $6$. This does not say that such subgroups must exist.
Also, a group $G$ of order $6$ having subgroups that are cyclic does NOT mean that the group $G$ itself is cyclic. In fact, all groups have cyclic subgroups by definition (and usually plenty of them).
If you think the statement is false, then your job is to find a group of order $6$ that is not cyclic. If you think it is true then you must prove that all such groups are cyclic. How many groups of order $6$ do you know?
Edit: never mind, answer given away in the comments.
$endgroup$
$begingroup$
Thank you for your comment - I will try to continue as I was clearly wrongly stuck in my own Lagrange theorem
$endgroup$
– winterdeur
Dec 3 '18 at 1:49
$begingroup$
I guess with order 2 I can proof that group G itself is cyclic.
$endgroup$
– winterdeur
Dec 3 '18 at 1:50
$begingroup$
To see a small example to illustrate your mistakes more simply, notice that all proper subgroups of $G=mathbb{Z}_2 times mathbb{Z}_2$ are cyclic, yet $G$ is not. (Your mistakes are also common, so don't worry too much now. You are right to try and fix them now.)
$endgroup$
– Randall
Dec 3 '18 at 1:51
$begingroup$
"I guess with order 2 I can proof that group G itself is cyclic. " Which gets you nowhere. The subgroup of order 2 is cyclic but that doesn't mean the group of which it is a subgroup is cyclic.
$endgroup$
– fleablood
Dec 3 '18 at 1:57
add a comment |
$begingroup$
To answer your question, it is not correct. Several mistakes here.
Lagrange does not say that such subgroups exist. You are incorrectly applying a false converse. Any subgroup of a group of order $6$ must have order $1, 2, 3$, or $6$. This does not say that such subgroups must exist.
Also, a group $G$ of order $6$ having subgroups that are cyclic does NOT mean that the group $G$ itself is cyclic. In fact, all groups have cyclic subgroups by definition (and usually plenty of them).
If you think the statement is false, then your job is to find a group of order $6$ that is not cyclic. If you think it is true then you must prove that all such groups are cyclic. How many groups of order $6$ do you know?
Edit: never mind, answer given away in the comments.
$endgroup$
To answer your question, it is not correct. Several mistakes here.
Lagrange does not say that such subgroups exist. You are incorrectly applying a false converse. Any subgroup of a group of order $6$ must have order $1, 2, 3$, or $6$. This does not say that such subgroups must exist.
Also, a group $G$ of order $6$ having subgroups that are cyclic does NOT mean that the group $G$ itself is cyclic. In fact, all groups have cyclic subgroups by definition (and usually plenty of them).
If you think the statement is false, then your job is to find a group of order $6$ that is not cyclic. If you think it is true then you must prove that all such groups are cyclic. How many groups of order $6$ do you know?
Edit: never mind, answer given away in the comments.
answered Dec 3 '18 at 1:47
RandallRandall
9,42111129
9,42111129
$begingroup$
Thank you for your comment - I will try to continue as I was clearly wrongly stuck in my own Lagrange theorem
$endgroup$
– winterdeur
Dec 3 '18 at 1:49
$begingroup$
I guess with order 2 I can proof that group G itself is cyclic.
$endgroup$
– winterdeur
Dec 3 '18 at 1:50
$begingroup$
To see a small example to illustrate your mistakes more simply, notice that all proper subgroups of $G=mathbb{Z}_2 times mathbb{Z}_2$ are cyclic, yet $G$ is not. (Your mistakes are also common, so don't worry too much now. You are right to try and fix them now.)
$endgroup$
– Randall
Dec 3 '18 at 1:51
$begingroup$
"I guess with order 2 I can proof that group G itself is cyclic. " Which gets you nowhere. The subgroup of order 2 is cyclic but that doesn't mean the group of which it is a subgroup is cyclic.
$endgroup$
– fleablood
Dec 3 '18 at 1:57
add a comment |
$begingroup$
Thank you for your comment - I will try to continue as I was clearly wrongly stuck in my own Lagrange theorem
$endgroup$
– winterdeur
Dec 3 '18 at 1:49
$begingroup$
I guess with order 2 I can proof that group G itself is cyclic.
$endgroup$
– winterdeur
Dec 3 '18 at 1:50
$begingroup$
To see a small example to illustrate your mistakes more simply, notice that all proper subgroups of $G=mathbb{Z}_2 times mathbb{Z}_2$ are cyclic, yet $G$ is not. (Your mistakes are also common, so don't worry too much now. You are right to try and fix them now.)
$endgroup$
– Randall
Dec 3 '18 at 1:51
$begingroup$
"I guess with order 2 I can proof that group G itself is cyclic. " Which gets you nowhere. The subgroup of order 2 is cyclic but that doesn't mean the group of which it is a subgroup is cyclic.
$endgroup$
– fleablood
Dec 3 '18 at 1:57
$begingroup$
Thank you for your comment - I will try to continue as I was clearly wrongly stuck in my own Lagrange theorem
$endgroup$
– winterdeur
Dec 3 '18 at 1:49
$begingroup$
Thank you for your comment - I will try to continue as I was clearly wrongly stuck in my own Lagrange theorem
$endgroup$
– winterdeur
Dec 3 '18 at 1:49
$begingroup$
I guess with order 2 I can proof that group G itself is cyclic.
$endgroup$
– winterdeur
Dec 3 '18 at 1:50
$begingroup$
I guess with order 2 I can proof that group G itself is cyclic.
$endgroup$
– winterdeur
Dec 3 '18 at 1:50
$begingroup$
To see a small example to illustrate your mistakes more simply, notice that all proper subgroups of $G=mathbb{Z}_2 times mathbb{Z}_2$ are cyclic, yet $G$ is not. (Your mistakes are also common, so don't worry too much now. You are right to try and fix them now.)
$endgroup$
– Randall
Dec 3 '18 at 1:51
$begingroup$
To see a small example to illustrate your mistakes more simply, notice that all proper subgroups of $G=mathbb{Z}_2 times mathbb{Z}_2$ are cyclic, yet $G$ is not. (Your mistakes are also common, so don't worry too much now. You are right to try and fix them now.)
$endgroup$
– Randall
Dec 3 '18 at 1:51
$begingroup$
"I guess with order 2 I can proof that group G itself is cyclic. " Which gets you nowhere. The subgroup of order 2 is cyclic but that doesn't mean the group of which it is a subgroup is cyclic.
$endgroup$
– fleablood
Dec 3 '18 at 1:57
$begingroup$
"I guess with order 2 I can proof that group G itself is cyclic. " Which gets you nowhere. The subgroup of order 2 is cyclic but that doesn't mean the group of which it is a subgroup is cyclic.
$endgroup$
– fleablood
Dec 3 '18 at 1:57
add a comment |
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$begingroup$
Not true, try $D_3$ the dihedral group
$endgroup$
– user25959
Dec 3 '18 at 1:47
$begingroup$
"he subgroup contains an element g of order 2 and the subgroup contains an element h of order 3." but that doesn't mean the order of $hg$ is $6$ or that any element has an order of $6$.
$endgroup$
– fleablood
Dec 3 '18 at 2:03