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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=

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<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>

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