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A note on extending partial automorphisms of abelian groups

Published online by Cambridge University Press:  09 April 2009

C. G. Chehata  and
A. Shawky
C. G. Chehata
Affiliation:
Faculty of Science The University of Alexandria, Egypt
A. Shawky
Affiliation:
Faculty of Science The University of Alexandria, Egypt
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Given a group G and a partial automorphism μ of G, i.e. an isomorphism mapping a subgroup A of G onto another subgroup B of G, then it is known [3] that μ can always be extended to a total automorphism, in fact an inner one, of a supergroup of G; that is there exists a group G* ⊇ G with an inner automorphism μ* whose effect on the elements of A is the same as that of μ. Also any number of partial automorphisms μσ, where a ranges over some index set Σ can be simultaneously extended to inner automorphisms of one and the same group [3, Theorem II].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 1 , February 1970 , pp. 37 - 41
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Chehata, C. G., ‘Commutative extension of partial automorphisms of groups’, Proc. Glasgow Math. Ass., I part IV (1953), 170–181.CrossRefGoogle Scholar
[2]Neumann, B. H. and Neumann, Hanna, ‘A remark on generalized free products’, J. London Math. Soc., 25 (1950), 202–204.CrossRefGoogle Scholar
[3]Higman, G., Neumann, B. H. and Neumann, Hanna, ‘Embedding theorems for groups’, J. London Math. Soc., 24 (1949), 247–254.CrossRefGoogle Scholar