On centrally orderable groups

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• 作者：V.V. Bludov ; A.M.W. Glass and A.H. Rhemtulla
• 刊名：Journal of Algebra
• 出版年：2005
• 期刊代码：69_00218693
• 类别：mt
• 出版时间：2005
• 卷：291
• 期：1
• 页码：129-143
• 文件大小：158 K

摘要

In b;V.V. Bludov, A.M.W. Glass, A.H. Rhemtulla, Ordered groups in which all convex jumps are central, J. Korean Math. Soc. 40 (2003) 225–239] we considered the class me="mml1">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml1&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=cb131dcb722336575808cfb681a44507">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-1/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=12 width=11> of all orderable groups with all orders having central convex jumps and the class me="mml2">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml2&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=3474cb190717fe0a4ed97dc3fda57992">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-4/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=15 width=17> of all orderable groups all of whose two generator subgroups belong to me="mml3">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml3&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=3ae446de0e0e8099b9aa3440b396fa4f">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-5/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=12 width=11>. Both these classes contain all locally nilpotent torsion-free groups. We proved that every soluble-by-finite group belonging to me="mml4">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml4&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=10d729015143d9253967b895c7ec4f6a">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-6/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=15 width=17> must be locally nilpotent, but there is a two-generator metabelian group belonging to me="mml5">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml5&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=828d24be8022796796bcc34d4dd5f049">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-7/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=15 width=40> (whence it is not locally nilpotent).

In this paper we consider the closure of me="mml6">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml6&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=ce73dc7a5669ca417332af22616de1f2">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-8/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=12 width=11> and me="mml7">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml7&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=25c692c3c3d5aa20611cb71245b1b63b">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-9/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=15 width=17> under elementary equivalence, direct sums and full Cartesian products, homomorphic images, and residual properties. We further show that every metabelian group G which has a central order can be embedded in a metabelian group me="mml8">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml8&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=0f1665c1db2cd7b29d5c025a93ca29f9">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-B/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=14 width=13> belonging to me="mml9">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml9&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=3786496ec5631ec3b935365ad2bd5da2">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-C/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=12 width=11> so that each central order on G can be extended to me="mml10">b=MathURL&_method=retrieve&_udi=B6WH2-4GG8W62-3&_mathId=mml10&_user=10&_cdi=6838&_rdoc=8&_handle=V-WA-A-W-WU-MsSAYVW-UUA-U-AABZWYBDWW-AABBYZVCWW-CADVYBEYA-WU-U&_acct=C000050221&_version=1&_userid=10&md5=da52543d9f7a79d23df920facb26a23a">mg src="http://www.sciencedirect.com/cache/MiamiImageURL/B6WH2-4GG8W62-3-2/0?wchp=dGLbVtz-zSkWb" alt="Click to view the MathML source" align="absbottom" border="0" height=14 width=13>. This is achieved by ascending HNN-extensions. We further show that a finitely generated metabelian group has a central order if and only if it is residually torsion-free nilpotent.