------=_NextPart_001_0011_01C14CDB.42F84440 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable
Dear GAP Forum,
I am doing research on groups. Specifically,
=E2=94=82Bn=E2=94=82 is the order of the group Bn. Bn is the collection =
of all subgroups of B with order n.
Our goal is to find all groups so that each =E2=94=82Bn=E2=94=82 will div=
ide the order of the group. We are looking at finite groups only.
For example, B =3D Z2 X Z3 has order 6.
There is 1 subgroup of order 2 : {(1,0)} -- 1 divides 6
There is 1 subgroup of order 3 : {(0,1)} -- 1 divides 6
There is 1 subgroup of order 6 : {(1,1)} -- 1 divides 6
so Z2 X Z3 has this property.
But Z2 X Z2 does not work. Z2 X Z2 has order 4.
There are 3 subgroups of order 3 : {(1,0), (0,1), (1,1)} -- 3 does NOT di=
vide 4
I have general cases for:
Zp X Zp X ... XZp where p is prime
Zp^n X Zp^n where p is prime and n is a natural number
Zp1p2 X Zp1p2 where p1 and p2 are distinct primes
I think I have general cases for others, but the sets are often too big t=
o do by hand and I was wondering if GAP had any way to find the possible =
subgroups and tell me their order so I can try to verify some of my other=
cases. If you need more clarification, feel free to ask. Thank you in =
advance for any help you are able to provide.
Chad T. Lower
a.k.a. Chopper
http://chadtlower.tripod.com/
Fight Spam! Join CAUCE! =3D=3D http://www.cauce.org/
Greet one another with the kiss of love.
Peace to all of you that are in Christ. I Peter 5:14
The suespammers.org mail server is located in California; do not send uns=
olicited bulk e-mail or unsolicited commercial e-mail to my suespammers.o=
------=_NextPart_001_0011_01C14CDB.42F84440 Content-Type: text/html; charset="utf-8" Content-Transfer-Encoding: quoted-printable
<HTML><BODY STYLE=3D"font:10pt verdana; border:none;"><DIV>Dear GAP Forum=
,</DIV> <DIV> </DIV> <DIV>I am doing research on groups. Speci=
fically,</DIV> <DIV> </DIV> <P class=3DMsoNormal style=3D"MARGIN: 0i=
n 0in 0pt"><FONT face=3D"Times New Roman" size=3D3>=E2=94=82B<SUB>n</SUB>=
=E2=94=82 is the order of the group B<SUB>n</SUB>.<SPAN style=3D"mso-spac=
erun: yes"> </SPAN>B<SUB>n</SUB> is the collection of all subgroups=
of B with order n.</FONT></P> <DIV><SPAN style=3D"FONT-SIZE: 12pt; COLOR=
: black; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times =
New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bi=
di-language: AR-SA">Our goal is to find all groups so that each =E2=94=82=
B<SUB>n</SUB>=E2=94=82 will divide the order of the group.<SPAN style=3D"=
mso-spacerun: yes"> </SPAN>We are looking at finite groups only.</S=
PAN></DIV> <DIV> </DIV> <DIV>For example, B =3D Z2 X Z3 has order 6.=
</DIV> <DIV>There is 1 subgroup of order 2 : {(1,0)} -- 1 divides 6</DIV>=
<DIV>There is 1 subgroup of order 3 : {(0,1)} -- 1 divides 6</DIV> <DIV>=
There is 1 subgroup of order 6 : {(1,1)} -- 1 divides 6</DIV> <DIV>so Z2 =
X Z3 has this property.</DIV> <DIV> </DIV> <DIV>But Z2 X Z2 does not=
work. Z2 X Z2 has order 4.</DIV> <DIV>There are 3 subgroups of ord=
er 3 : {(1,0), (0,1), (1,1)} -- 3 does NOT divide 4</DIV> <DIV> </DI=
V> <DIV>I have general cases for:</DIV> <DIV> </DIV> <DIV>Zp X Zp X =
... XZp where p is prime</DIV> <DIV>Zp^n X Zp^n where p is prime and n is=
a natural number</DIV> <DIV>Zp1p2 X Zp1p2 where p1 and p2 are distinct p=
rimes</DIV> <DIV> </DIV> <DIV>I think I have general cases for other=
s, but the sets are often too big to do by hand and I was wondering if GA=
P had any way to find the possible subgroups and tell me their order so I=
can try to verify some of my other cases. If you need more clarifi=
cation, feel free to ask. Thank you in advance for any help you are=
able to provide.<BR><BR>Chad T. Lower<BR>a.k.a. Chopper<BR>http://chadtl=
ower.tripod.com/<BR><BR>Fight Spam! Join CAUCE! =3D=3D http://www.cauce.o=
rg/<BR><BR>Greet one another with the kiss of love.<BR>Peace to all of yo=
u that are in Christ. I Peter 5:14<BR><BR>The suespammers.org mail server=
is located in California; do not send unsolicited bulk e-mail or unsolic=
ited commercial e-mail to my suespammers.org address.</DIV></BODY></HTML>=
<br clear=3Dall><hr>Get more from the Web. FREE MSN Explorer download : =
<a href=3D'http://explorer.msn.com'>http://explorer.msn.com</a><br></p>
------=_NextPart_001_0011_01C14CDB.42F84440--
Miles-Receive-Header: reply