Show that $D_11 oplus mathbbZ_3 not cong D_3 oplus mathbbZ_11$ Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Center of dihedral groupShow that $D_33$ is not isomorphic to $D_11 oplus Z_3$.Explain why $U_44 cong (mathbbZ_10 oplus mathbbZ_2) $.Prove that $D_3 oplus D_4$ is not isomorphic to $D_12oplusmathbb Z_2$Converse of sylow theoremShow $D_3cong S_3$ and $D_nncong S_n$ for $ngt 3$Proof that $S_3$ isomorphic to $D_3$$D_6 cong D_3times mathbb Z_2$Show that $ mathbbZ_6 oplus mathbbZ_6/ langle (2,3) rangle $ is or is not cyclic.Is $mathbbZ_84 oplus mathbbZ_72$ isomorphic to $mathbbZ_36 oplus mathbbZ_168$?How to show $D_3oplus D_4$ is not isomorphic to $D_24$?Show that $D_33$ is not isomorphic to $D_11 oplus Z_3$.
Strange behaviour of Check
Geometric mean and geometric standard deviation
What to do with post with dry rot?
Active filter with series inductor and resistor - do these exist?
What was the last x86 CPU that did not have the x87 floating-point unit built in?
What are the performance impacts of 'functional' Rust?
Statistical model of ligand substitution
How to market an anarchic city as a tourism spot to people living in civilized areas?
What computer would be fastest for Mathematica Home Edition?
Can a monk deflect thrown melee weapons?
What items from the Roman-age tech-level could be used to deter all creatures from entering a small area?
Passing functions in C++
Direct Experience of Meditation
Notation for two qubit composite product state
Antler Helmet: Can it work?
How do I automatically answer y in bash script?
Single author papers against my advisor's will?
Why don't the Weasley twins use magic outside of school if the Trace can only find the location of spells cast?
If I can make up priors, why can't I make up posteriors?
Unable to start mainnet node docker container
What do you call a plan that's an alternative plan in case your initial plan fails?
Estimate capacitor parameters
If A makes B more likely then B makes A more likely"
Fishing simulator
Show that $D_11 oplus mathbbZ_3 not cong D_3 oplus mathbbZ_11$
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Center of dihedral groupShow that $D_33$ is not isomorphic to $D_11 oplus Z_3$.Explain why $U_44 cong (mathbbZ_10 oplus mathbbZ_2) $.Prove that $D_3 oplus D_4$ is not isomorphic to $D_12oplusmathbb Z_2$Converse of sylow theoremShow $D_3cong S_3$ and $D_nncong S_n$ for $ngt 3$Proof that $S_3$ isomorphic to $D_3$$D_6 cong D_3times mathbb Z_2$Show that $ mathbbZ_6 oplus mathbbZ_6/ langle (2,3) rangle $ is or is not cyclic.Is $mathbbZ_84 oplus mathbbZ_72$ isomorphic to $mathbbZ_36 oplus mathbbZ_168$?How to show $D_3oplus D_4$ is not isomorphic to $D_24$?Show that $D_33$ is not isomorphic to $D_11 oplus Z_3$.
$begingroup$
Show that $D_11 oplus mathbbZ_3 not cong D_3 oplus mathbbZ_11$
The first thing I check is that the size of both groups is the same since
$|D_11|=22$ and $|mathbbZ_3|=3$ thus $|D_11 oplus mathbbZ_3=22cdot 3=66$ and likewise $|D_3|=6$ and $|mathbbZ_11|=11$ thus $|D_3 oplus mathbbZ_11|=6cdot 11=66$
Could I evaluate the order of the centers of each group to show they are not the same size thus not isomoprhic? I see that the $|Z(D_11 oplus mathbbZ_3)|not = |Z(D_3 oplus mathbbZ_11)|$ since the order of both center of the dihedral groups is only the identity where as the order of the center of $mathbbZ_3$ is not the same size as the order of the center of $mathbbZ_11$
group-theory finite-groups
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
$endgroup$
add a comment |
$begingroup$
Show that $D_11 oplus mathbbZ_3 not cong D_3 oplus mathbbZ_11$
The first thing I check is that the size of both groups is the same since
$|D_11|=22$ and $|mathbbZ_3|=3$ thus $|D_11 oplus mathbbZ_3=22cdot 3=66$ and likewise $|D_3|=6$ and $|mathbbZ_11|=11$ thus $|D_3 oplus mathbbZ_11|=6cdot 11=66$
Could I evaluate the order of the centers of each group to show they are not the same size thus not isomoprhic? I see that the $|Z(D_11 oplus mathbbZ_3)|not = |Z(D_3 oplus mathbbZ_11)|$ since the order of both center of the dihedral groups is only the identity where as the order of the center of $mathbbZ_3$ is not the same size as the order of the center of $mathbbZ_11$
group-theory finite-groups
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
$endgroup$
$begingroup$
Elements of order 2
$endgroup$
– Randall
Dec 11 '17 at 18:49
4
$begingroup$
Yes, the fact that their centres have different orders shows that they are not isomorphic.
$endgroup$
– Derek Holt
Dec 11 '17 at 18:51
$begingroup$
There are lots of other approaches as well: For example, In one of these groups there's an element of order 11 that fails to commute with some other element. Is there in the other? It's worth thinking about other possibilities to try to build up your toolbox of skills.
$endgroup$
– John Hughes
Dec 11 '17 at 18:53
add a comment |
$begingroup$
Show that $D_11 oplus mathbbZ_3 not cong D_3 oplus mathbbZ_11$
The first thing I check is that the size of both groups is the same since
$|D_11|=22$ and $|mathbbZ_3|=3$ thus $|D_11 oplus mathbbZ_3=22cdot 3=66$ and likewise $|D_3|=6$ and $|mathbbZ_11|=11$ thus $|D_3 oplus mathbbZ_11|=6cdot 11=66$
Could I evaluate the order of the centers of each group to show they are not the same size thus not isomoprhic? I see that the $|Z(D_11 oplus mathbbZ_3)|not = |Z(D_3 oplus mathbbZ_11)|$ since the order of both center of the dihedral groups is only the identity where as the order of the center of $mathbbZ_3$ is not the same size as the order of the center of $mathbbZ_11$
group-theory finite-groups
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
$endgroup$
Show that $D_11 oplus mathbbZ_3 not cong D_3 oplus mathbbZ_11$
The first thing I check is that the size of both groups is the same since
$|D_11|=22$ and $|mathbbZ_3|=3$ thus $|D_11 oplus mathbbZ_3=22cdot 3=66$ and likewise $|D_3|=6$ and $|mathbbZ_11|=11$ thus $|D_3 oplus mathbbZ_11|=6cdot 11=66$
Could I evaluate the order of the centers of each group to show they are not the same size thus not isomoprhic? I see that the $|Z(D_11 oplus mathbbZ_3)|not = |Z(D_3 oplus mathbbZ_11)|$ since the order of both center of the dihedral groups is only the identity where as the order of the center of $mathbbZ_3$ is not the same size as the order of the center of $mathbbZ_11$
group-theory finite-groups
group-theory finite-groups
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
asked Dec 11 '17 at 18:45
HighSchool15HighSchool15
1,074519
1,074519
$begingroup$
Elements of order 2
$endgroup$
– Randall
Dec 11 '17 at 18:49
4
$begingroup$
Yes, the fact that their centres have different orders shows that they are not isomorphic.
$endgroup$
– Derek Holt
Dec 11 '17 at 18:51
$begingroup$
There are lots of other approaches as well: For example, In one of these groups there's an element of order 11 that fails to commute with some other element. Is there in the other? It's worth thinking about other possibilities to try to build up your toolbox of skills.
$endgroup$
– John Hughes
Dec 11 '17 at 18:53
add a comment |
$begingroup$
Elements of order 2
$endgroup$
– Randall
Dec 11 '17 at 18:49
4
$begingroup$
Yes, the fact that their centres have different orders shows that they are not isomorphic.
$endgroup$
– Derek Holt
Dec 11 '17 at 18:51
$begingroup$
There are lots of other approaches as well: For example, In one of these groups there's an element of order 11 that fails to commute with some other element. Is there in the other? It's worth thinking about other possibilities to try to build up your toolbox of skills.
$endgroup$
– John Hughes
Dec 11 '17 at 18:53
$begingroup$
Elements of order 2
$endgroup$
– Randall
Dec 11 '17 at 18:49
$begingroup$
Elements of order 2
$endgroup$
– Randall
Dec 11 '17 at 18:49
4
4
$begingroup$
Yes, the fact that their centres have different orders shows that they are not isomorphic.
$endgroup$
– Derek Holt
Dec 11 '17 at 18:51
$begingroup$
Yes, the fact that their centres have different orders shows that they are not isomorphic.
$endgroup$
– Derek Holt
Dec 11 '17 at 18:51
$begingroup$
There are lots of other approaches as well: For example, In one of these groups there's an element of order 11 that fails to commute with some other element. Is there in the other? It's worth thinking about other possibilities to try to build up your toolbox of skills.
$endgroup$
– John Hughes
Dec 11 '17 at 18:53
$begingroup$
There are lots of other approaches as well: For example, In one of these groups there's an element of order 11 that fails to commute with some other element. Is there in the other? It's worth thinking about other possibilities to try to build up your toolbox of skills.
$endgroup$
– John Hughes
Dec 11 '17 at 18:53
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As noted the center of both groups is different, hence the groups cannot be isomorphic. We have $Z(D_2n+1)=1$ for all $nge 1$ and thus
$$
Z(D_11times C_3)=Z(D_11)times Z(C_3)=1times C_3cong C_3,
$$
and
$$
Z(D_3times C_11)=Z(D_3)times Z(C_11)=1times C_11cong C_11.
$$
Reference: Center of dihedral group
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2562091%2fshow-that-d-11-oplus-mathbbz-3-not-cong-d-3-oplus-mathbbz-11%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As noted the center of both groups is different, hence the groups cannot be isomorphic. We have $Z(D_2n+1)=1$ for all $nge 1$ and thus
$$
Z(D_11times C_3)=Z(D_11)times Z(C_3)=1times C_3cong C_3,
$$
and
$$
Z(D_3times C_11)=Z(D_3)times Z(C_11)=1times C_11cong C_11.
$$
Reference: Center of dihedral group
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
$endgroup$
add a comment |
$begingroup$
As noted the center of both groups is different, hence the groups cannot be isomorphic. We have $Z(D_2n+1)=1$ for all $nge 1$ and thus
$$
Z(D_11times C_3)=Z(D_11)times Z(C_3)=1times C_3cong C_3,
$$
and
$$
Z(D_3times C_11)=Z(D_3)times Z(C_11)=1times C_11cong C_11.
$$
Reference: Center of dihedral group
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
$endgroup$
add a comment |
$begingroup$
As noted the center of both groups is different, hence the groups cannot be isomorphic. We have $Z(D_2n+1)=1$ for all $nge 1$ and thus
$$
Z(D_11times C_3)=Z(D_11)times Z(C_3)=1times C_3cong C_3,
$$
and
$$
Z(D_3times C_11)=Z(D_3)times Z(C_11)=1times C_11cong C_11.
$$
Reference: Center of dihedral group
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
$endgroup$
As noted the center of both groups is different, hence the groups cannot be isomorphic. We have $Z(D_2n+1)=1$ for all $nge 1$ and thus
$$
Z(D_11times C_3)=Z(D_11)times Z(C_3)=1times C_3cong C_3,
$$
and
$$
Z(D_3times C_11)=Z(D_3)times Z(C_11)=1times C_11cong C_11.
$$
Reference: Center of dihedral group
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
answered Mar 25 at 19:58
Dietrich BurdeDietrich Burde
82k649107
82k649107
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2562091%2fshow-that-d-11-oplus-mathbbz-3-not-cong-d-3-oplus-mathbbz-11%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Elements of order 2
$endgroup$
– Randall
Dec 11 '17 at 18:49
4
$begingroup$
Yes, the fact that their centres have different orders shows that they are not isomorphic.
$endgroup$
– Derek Holt
Dec 11 '17 at 18:51
$begingroup$
There are lots of other approaches as well: For example, In one of these groups there's an element of order 11 that fails to commute with some other element. Is there in the other? It's worth thinking about other possibilities to try to build up your toolbox of skills.
$endgroup$
– John Hughes
Dec 11 '17 at 18:53