$Z(G)=left langle e right rangle$ if and only if $N_Gamma...
Let $G$ a group and let a group $Gamma=Gtimes G$ with $forall (a,b),(c,d)inGamma, (a,b)(c,d):=(ac,bd)$. Let a map $varphi:GtoGamma$ by $varphi(g)=(g,g)$. Prove that $Z(G)=left langle e right rangle$ if and only if $N_Gamma (Im(varphi))=Im(varphi)$.
Define $Q:=operatorname{Im}(varphi)$. I've shown that $Q={(g,g):gin G}$, $varphi$ is a homomorphism and that $G$ is an abelian group if and only if $Q$ is a normal subgroup of $Gamma$.
abstract-algebra group-theory group-homomorphism
edited Nov 28 '18 at 19:36
Bernard
118k639112
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
closed as off-topic by Derek Holt, amWhy, Leucippus, KReiser, max_zorn Nov 29 '18 at 6:24
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Let $G$ a group and let a group $Gamma=Gtimes G$ with $forall (a,b),(c,d)inGamma, (a,b)(c,d):=(ac,bd)$. Let a map $varphi:GtoGamma$ by $varphi(g)=(g,g)$. Prove that $Z(G)=left langle e right rangle$ if and only if $N_Gamma (Im(varphi))=Im(varphi)$.
Define $Q:=operatorname{Im}(varphi)$. I've shown that $Q={(g,g):gin G}$, $varphi$ is a homomorphism and that $G$ is an abelian group if and only if $Q$ is a normal subgroup of $Gamma$.
abstract-algebra group-theory group-homomorphism
edited Nov 28 '18 at 19:36
Bernard
118k639112
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
closed as off-topic by Derek Holt, amWhy, Leucippus, KReiser, max_zorn Nov 29 '18 at 6:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Derek Holt, amWhy, Leucippus, KReiser, max_zorn
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
Let $G$ a group and let a group $Gamma=Gtimes G$ with $forall (a,b),(c,d)inGamma, (a,b)(c,d):=(ac,bd)$. Let a map $varphi:GtoGamma$ by $varphi(g)=(g,g)$. Prove that $Z(G)=left langle e right rangle$ if and only if $N_Gamma (Im(varphi))=Im(varphi)$.
Define $Q:=operatorname{Im}(varphi)$. I've shown that $Q={(g,g):gin G}$, $varphi$ is a homomorphism and that $G$ is an abelian group if and only if $Q$ is a normal subgroup of $Gamma$.
abstract-algebra group-theory group-homomorphism
edited Nov 28 '18 at 19:36
Bernard
118k639112
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
Let $G$ a group and let a group $Gamma=Gtimes G$ with $forall (a,b),(c,d)inGamma, (a,b)(c,d):=(ac,bd)$. Let a map $varphi:GtoGamma$ by $varphi(g)=(g,g)$. Prove that $Z(G)=left langle e right rangle$ if and only if $N_Gamma (Im(varphi))=Im(varphi)$.
Define $Q:=operatorname{Im}(varphi)$. I've shown that $Q={(g,g):gin G}$, $varphi$ is a homomorphism and that $G$ is an abelian group if and only if $Q$ is a normal subgroup of $Gamma$.
abstract-algebra group-theory group-homomorphism
abstract-algebra group-theory group-homomorphism
edited Nov 28 '18 at 19:36
Bernard
118k639112
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
edited Nov 28 '18 at 19:36
Bernard
118k639112
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
edited Nov 28 '18 at 19:36
Bernard
118k639112
edited Nov 28 '18 at 19:36
Bernard
118k639112
edited Nov 28 '18 at 19:36
Bernard
118k639112
118k639112
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
asked Nov 28 '18 at 17:32
J. DoeJ. Doe
1286
1286
closed as off-topic by Derek Holt, amWhy, Leucippus, KReiser, max_zorn Nov 29 '18 at 6:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Derek Holt, amWhy, Leucippus, KReiser, max_zorn
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Derek Holt, amWhy, Leucippus, KReiser, max_zorn Nov 29 '18 at 6:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Derek Holt, amWhy, Leucippus, KReiser, max_zorn
If this question can be reworded to fit the rules in the help center, please edit the question.
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Suppose $Z(G)=langle erangle$ and let $(a,b)in N_Gamma(operatorname{Im}(phi))$. Then for all $gin G$ we have
$$operatorname{Im}(phi)ni (a,b)(g,g)(a,b)^{-1}=(aga^{-1},bgb^{-1}),$$
which (by your calculation of $operatorname{Im}(phi)$) means that for some $g'in G$ we have
$$(aga^{-1},bgb^{-1})=(g',g')implies aga^{-1}=bgb^{-1}.$$
Rearrange this equation to see that $a^{-1}b$ commutes with $g$, and since this holds for every $gin G$ we have $a^{-1}bin Z(G)=langle erangle$, which implies $a=b$ so $(a,b)in operatorname{Im}(phi)$.
edited Nov 28 '18 at 19:37
Bernard
118k639112
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Suppose $Z(G)=langle erangle$ and let $(a,b)in N_Gamma(operatorname{Im}(phi))$. Then for all $gin G$ we have
$$operatorname{Im}(phi)ni (a,b)(g,g)(a,b)^{-1}=(aga^{-1},bgb^{-1}),$$
which (by your calculation of $operatorname{Im}(phi)$) means that for some $g'in G$ we have
$$(aga^{-1},bgb^{-1})=(g',g')implies aga^{-1}=bgb^{-1}.$$
Rearrange this equation to see that $a^{-1}b$ commutes with $g$, and since this holds for every $gin G$ we have $a^{-1}bin Z(G)=langle erangle$, which implies $a=b$ so $(a,b)in operatorname{Im}(phi)$.
edited Nov 28 '18 at 19:37
Bernard
118k639112
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
add a comment |
Suppose $Z(G)=langle erangle$ and let $(a,b)in N_Gamma(operatorname{Im}(phi))$. Then for all $gin G$ we have
$$operatorname{Im}(phi)ni (a,b)(g,g)(a,b)^{-1}=(aga^{-1},bgb^{-1}),$$
which (by your calculation of $operatorname{Im}(phi)$) means that for some $g'in G$ we have
$$(aga^{-1},bgb^{-1})=(g',g')implies aga^{-1}=bgb^{-1}.$$
Rearrange this equation to see that $a^{-1}b$ commutes with $g$, and since this holds for every $gin G$ we have $a^{-1}bin Z(G)=langle erangle$, which implies $a=b$ so $(a,b)in operatorname{Im}(phi)$.
edited Nov 28 '18 at 19:37
Bernard
118k639112
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
add a comment |
Suppose $Z(G)=langle erangle$ and let $(a,b)in N_Gamma(operatorname{Im}(phi))$. Then for all $gin G$ we have
$$operatorname{Im}(phi)ni (a,b)(g,g)(a,b)^{-1}=(aga^{-1},bgb^{-1}),$$
which (by your calculation of $operatorname{Im}(phi)$) means that for some $g'in G$ we have
$$(aga^{-1},bgb^{-1})=(g',g')implies aga^{-1}=bgb^{-1}.$$
Rearrange this equation to see that $a^{-1}b$ commutes with $g$, and since this holds for every $gin G$ we have $a^{-1}bin Z(G)=langle erangle$, which implies $a=b$ so $(a,b)in operatorname{Im}(phi)$.
edited Nov 28 '18 at 19:37
Bernard
118k639112
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
Suppose $Z(G)=langle erangle$ and let $(a,b)in N_Gamma(operatorname{Im}(phi))$. Then for all $gin G$ we have
$$operatorname{Im}(phi)ni (a,b)(g,g)(a,b)^{-1}=(aga^{-1},bgb^{-1}),$$
which (by your calculation of $operatorname{Im}(phi)$) means that for some $g'in G$ we have
$$(aga^{-1},bgb^{-1})=(g',g')implies aga^{-1}=bgb^{-1}.$$
Rearrange this equation to see that $a^{-1}b$ commutes with $g$, and since this holds for every $gin G$ we have $a^{-1}bin Z(G)=langle erangle$, which implies $a=b$ so $(a,b)in operatorname{Im}(phi)$.
edited Nov 28 '18 at 19:37
Bernard
118k639112
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
edited Nov 28 '18 at 19:37
Bernard
118k639112
edited Nov 28 '18 at 19:37
Bernard
118k639112
edited Nov 28 '18 at 19:37
Bernard
118k639112
118k639112
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
answered Nov 28 '18 at 17:44
Alex MathersAlex Mathers
10.8k21344
10.8k21344
add a comment |
add a comment |
