Generalized direct products of groups and their application to the study of residuality of free constructions of groups

In this paper, by establishing a new graph ?(G) over the semi-direct product of groups, we will first state and prove some graph-theoretical properties, namely, diameter, maximum and minimum degrees, girth, degree sequence, domination number, chromatic number, clique number of ?(G). In the final section we will show that ?(G) is actually a perfect graph.

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A Note on Generalized Direct Products of Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1973-010-2 ◽

1973 ◽

Vol 25 (1) ◽

pp. 115-116

Author(s):

Marlene Schick

Keyword(s):

Direct Product ◽

Direct Products ◽

Compatibility Conditions ◽

Cyclic Groups ◽

Products Of Groups ◽

Finite Set

In [1] Tang proved that the generalized direct product of a finite set of cyclic groups amalgamating subgroups which satisfy certain compatibility conditions always exists. In the proof, Theorem 4.1 is made use of. However, this theorem is not correct since we can construct examples of groups which satisfy the conditions of Theorem 4.1, but whose generalized direct product does not exist. Therefore, a modification of this result as pointed out by Professor Tang is given here, together with the resulting modification of the proof of the result stated above.

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On Hopf Algebra Extensions Arising from Semi-Direct Products of Groups

Groups, Rings, Lie and Hopf Algebras ◽

10.1007/978-1-4613-0235-3_9 ◽

2003 ◽

pp. 129-139 ◽

Cited By ~ 1

Author(s):

Mitja Mastnak

Keyword(s):

Hopf Algebra ◽

Direct Products ◽

Products Of Groups

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K. A. Mihajlova. Probléma vhoždéniá dlá prámyh proizvédénij grupp (The occurrence problem for direct products of groups). Doklady Akadémii Nauk SSSR, vol. 119 (1958), pp. 1103–1105. - K. A. Mihajlova. Probléma vhoždéniá dlá prámyh proizvédénij grupp (The occurrence problem for direct products of groups). Matématičéskij sbornik, n.s. vol. 70 (1966), pp. 241–251.

Journal of Symbolic Logic ◽

10.2307/2270002 ◽

1971 ◽

Vol 36 (3) ◽

pp. 540-541

Author(s):

Donald J. Collins

Keyword(s):

Nauk Sssr ◽

Direct Products ◽

Akademii Nauk Sssr ◽

Doklady Akademii Nauk Sssr ◽

Occurrence Problem ◽

Products Of Groups

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Presentations for some direct products of groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700026186 ◽

1983 ◽

Vol 28 (1) ◽

pp. 131-133 ◽

Cited By ~ 5

Author(s):

P.E. Kenne

Keyword(s):

Direct Product ◽

Alternating Group ◽

Direct Products ◽

Icosahedral Group ◽

Products Of Groups

We give efficient presentations for the direct product of two copies of the alternating group of degree five and the direct product of the alternating group of degree five and the binary icosahedral group.

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