Is $U_8$ isomorphic to $K_4$ ( Klein Group)
$U_8=1,3,5,7$ since this group has one element of order one, three elements of two order and no element of $4$ order .. so does the Klein group.
Both $U(8)$ and the Klein group are non cyclic groups whose every proper subgroup is cyclic, so the Klein group is isomorphic to U(8)?
abstract-algebra group-theory finite-groups group-isomorphism
edited Dec 26 at 13:31
amWhy
191k28224439
asked Dec 26 at 13:08
Henry
325
add a comment |
$U_8=1,3,5,7$ since this group has one element of order one, three elements of two order and no element of $4$ order .. so does the Klein group.
Both $U(8)$ and the Klein group are non cyclic groups whose every proper subgroup is cyclic, so the Klein group is isomorphic to U(8)?
abstract-algebra group-theory finite-groups group-isomorphism
edited Dec 26 at 13:31
amWhy
191k28224439
asked Dec 26 at 13:08
Henry
325
is this the isomorphism defined](i.stack.imgur.com/HZTuj.jpg)](https://i.stack.imgur.com/…
– Henry
Dec 26 at 13:10
See also this question, and this one.
– Dietrich Burde
Dec 26 at 15:51
add a comment |
$U_8=1,3,5,7$ since this group has one element of order one, three elements of two order and no element of $4$ order .. so does the Klein group.
Both $U(8)$ and the Klein group are non cyclic groups whose every proper subgroup is cyclic, so the Klein group is isomorphic to U(8)?
abstract-algebra group-theory finite-groups group-isomorphism
edited Dec 26 at 13:31
amWhy
191k28224439
asked Dec 26 at 13:08
Henry
325
$U_8=1,3,5,7$ since this group has one element of order one, three elements of two order and no element of $4$ order .. so does the Klein group.
Both $U(8)$ and the Klein group are non cyclic groups whose every proper subgroup is cyclic, so the Klein group is isomorphic to U(8)?
abstract-algebra group-theory finite-groups group-isomorphism
abstract-algebra group-theory finite-groups group-isomorphism
edited Dec 26 at 13:31
amWhy
191k28224439
asked Dec 26 at 13:08
Henry
325
edited Dec 26 at 13:31
amWhy
191k28224439
asked Dec 26 at 13:08
Henry
325
edited Dec 26 at 13:31
amWhy
191k28224439
edited Dec 26 at 13:31
amWhy
191k28224439
edited Dec 26 at 13:31
amWhy
191k28224439
191k28224439
asked Dec 26 at 13:08
Henry
325
asked Dec 26 at 13:08
Henry
325
asked Dec 26 at 13:08
Henry
325
325
is this the isomorphism defined](i.stack.imgur.com/HZTuj.jpg)](https://i.stack.imgur.com/…
– Henry
Dec 26 at 13:10
See also this question, and this one.
– Dietrich Burde
Dec 26 at 15:51
add a comment |
is this the isomorphism defined](i.stack.imgur.com/HZTuj.jpg)](https://i.stack.imgur.com/…
– Henry
Dec 26 at 13:10
See also this question, and this one.
– Dietrich Burde
Dec 26 at 15:51
is this the isomorphism defined](i.stack.imgur.com/HZTuj.jpg)](https://i.stack.imgur.com/…
– Henry
Dec 26 at 13:10
is this the isomorphism defined](i.stack.imgur.com/HZTuj.jpg)](https://i.stack.imgur.com/…
– Henry
Dec 26 at 13:10
See also this question, and this one.
– Dietrich Burde
Dec 26 at 15:51
See also this question, and this one.
– Dietrich Burde
Dec 26 at 15:51
add a comment |
1 Answer
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There is only two groups of order four: (1) the cyclic group and (2) the Klein group.
As all elements of $U(8)$ are of order $2$, $U(8)$ is indeed isomorphic as a group to the Klein group.
The key argument is that there is no other groups of order four than the two mentioned above
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
hey can u help finding all proper subgroups of Z2 × Z2 × Z2 ? i want to know how to approach to such problems ..any methodol
– Henry
Dec 26 at 13:27
I suggest you post another question to avoid mixing different topics.
– mathcounterexamples.net
Dec 26 at 13:28
add a comment |
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There is only two groups of order four: (1) the cyclic group and (2) the Klein group.
As all elements of $U(8)$ are of order $2$, $U(8)$ is indeed isomorphic as a group to the Klein group.
The key argument is that there is no other groups of order four than the two mentioned above
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
hey can u help finding all proper subgroups of Z2 × Z2 × Z2 ? i want to know how to approach to such problems ..any methodol
– Henry
Dec 26 at 13:27
I suggest you post another question to avoid mixing different topics.
– mathcounterexamples.net
Dec 26 at 13:28
add a comment |
There is only two groups of order four: (1) the cyclic group and (2) the Klein group.
As all elements of $U(8)$ are of order $2$, $U(8)$ is indeed isomorphic as a group to the Klein group.
The key argument is that there is no other groups of order four than the two mentioned above
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
hey can u help finding all proper subgroups of Z2 × Z2 × Z2 ? i want to know how to approach to such problems ..any methodol
– Henry
Dec 26 at 13:27
I suggest you post another question to avoid mixing different topics.
– mathcounterexamples.net
Dec 26 at 13:28
add a comment |
There is only two groups of order four: (1) the cyclic group and (2) the Klein group.
As all elements of $U(8)$ are of order $2$, $U(8)$ is indeed isomorphic as a group to the Klein group.
The key argument is that there is no other groups of order four than the two mentioned above
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
There is only two groups of order four: (1) the cyclic group and (2) the Klein group.
As all elements of $U(8)$ are of order $2$, $U(8)$ is indeed isomorphic as a group to the Klein group.
The key argument is that there is no other groups of order four than the two mentioned above
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
answered Dec 26 at 13:13
mathcounterexamples.net
24.3k21753
24.3k21753
hey can u help finding all proper subgroups of Z2 × Z2 × Z2 ? i want to know how to approach to such problems ..any methodol
– Henry
Dec 26 at 13:27
I suggest you post another question to avoid mixing different topics.
– mathcounterexamples.net
Dec 26 at 13:28
add a comment |
hey can u help finding all proper subgroups of Z2 × Z2 × Z2 ? i want to know how to approach to such problems ..any methodol
– Henry
Dec 26 at 13:27
I suggest you post another question to avoid mixing different topics.
– mathcounterexamples.net
Dec 26 at 13:28
hey can u help finding all proper subgroups of Z2 × Z2 × Z2 ? i want to know how to approach to such problems ..any methodol
– Henry
Dec 26 at 13:27
hey can u help finding all proper subgroups of Z2 × Z2 × Z2 ? i want to know how to approach to such problems ..any methodol
– Henry
Dec 26 at 13:27
I suggest you post another question to avoid mixing different topics.
– mathcounterexamples.net
Dec 26 at 13:28
I suggest you post another question to avoid mixing different topics.
– mathcounterexamples.net
Dec 26 at 13:28
add a comment |
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is this the isomorphism defined](i.stack.imgur.com/HZTuj.jpg)](https://i.stack.imgur.com/…
– Henry
Dec 26 at 13:10
See also this question, and this one.
– Dietrich Burde
Dec 26 at 15:51