Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-07-07T13:47:51.324Z Has data issue: false hasContentIssue false
  • English
  • Français

On Permutational Products of Groups:Part 2-amalgamated products

Published online by Cambridge University Press:  09 April 2009

R. J. Gregorac
R. J. Gregorac
Affiliation:
Institute of Advanced StudiesAustralian National University, Canberra
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The standard methods of constructing generalized free products of groups (with a single amalgamated subgroup) and permutational products of groups are to consider groups of permutations on sets. Although there is an apparent similarity between these two constructions, the exact nature of the relationship is not clear. The following addendum to [4] grew out of an attempt to determine this relationship. By noting that the original construction of permutational products (B. H. Neumann [7]) deals with a group of permutations on a group (although the group structure has been previously ignored; see [7], [8]) we here give an extension of the original permutational product-construction which yields both the generalized free product and the permutational products as groups of permutations on groups. A generalized free product is represented as a group of permutations on the ordinary free product of the constituents of the underlying group amalgam and a permutational product is a group of permutations on the direct product of the constituents of the amalgam.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1971 , pp. 21 - 34
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Allenby, R. B. J. T., ‘Normal forms for generalised regular products of groups’, Math. Zeitschr. 102 (1967) 356–369.CrossRefGoogle Scholar
[2]Allenby, R. B. J. T., ‘Permutational Products and Regular Products of Groups’, Trans. Amer. Math. Soc. 13 (1969) 537–543.CrossRefGoogle Scholar
[3]Golovin, O. N., ‘Nilpotent products of groups’, Mat. Sb. 27 (69) (1950), 427–454 Amer. Math. Soc. Transl. Ser. II 2 (1956), 89–115.Google Scholar
[4]Gregorac, R. J., ‘On Permutational Products of Groups’, J. Austral. Math. Soc., 10 (1969), 111–135.CrossRefGoogle Scholar
[5]Moran, S., ‘Associative operations on groups, I’, Proc. London Math. Soc. (3) 6 (1956), 581–596.CrossRefGoogle Scholar
[6]Moran, S., ‘Associative operations on groups, II’, Proc. London Math. Soc. (3) 8 (1958), 548–568.CrossRefGoogle Scholar
[7]Neumann, B. H., ‘An essay on free products of groups with amalgamations’ Phil. Trans. Royal Soc. of London (A) 246 (1954), 503–554.Google Scholar
[8]Neumann, B. H., ‘Permutational Products of Groups’, J. Austral. Math. Soc. 1, (1960) 299–310.CrossRefGoogle Scholar
[9]Neumann, B. H., ‘On Amalgams of Periodic Groups’, Proc. Roy. Soc. (A) 255 (1960), 477–489.Google Scholar
[10]Šmel'kin, A. L., ‘Nilpotent products and torsion-free nilpotent groups’, Sibirsk. Math. Journal, 3, No. 4 (1962) 625–640.Google Scholar
[11]Wiegold, J., ‘Nilpotent products of groups with amalgamations’, Publ. Math. Debrecen 6 (1959), 131–168.CrossRefGoogle Scholar
[12]Wiegold, J., ‘Some remarks on generalised products of groups with amalgamations’, Math. Zeitschr. 75 (1961), 57–78.CrossRefGoogle Scholar