Non abelian exterior square of a group.

Question

The non-abelian exterior square $G\wedge G$ of a group $G$ is defined as a group generated by the words $g\wedge h$, $g,h \in G$ related to the conditions

1. $g\wedge g=1$
2. $(g\wedge h)(h\wedge g)=1$
3. $g\wedge g'h=(g\wedge g').(^{g'}g\wedge ^{g'}h)$
4. $gg'\wedge h=(^gg'\wedge ^gh).(g\wedge h)$

where $^gh=ghg^{-1}$. The non abelian tensor square $G \otimes G$ of a group $G$ is determined by many authors For the groups of order $p^2q$ it is computed. KIndly give me some idea or refrence, so I can compute it for some groups.

Answer 1

The standard notation for the nonabelian exterior square is $G\wedge G$ (just like the notation for the nonabelian tensor square is $G\otimes G$). The notation $G*G$ is commonly understood to represent the free product of $G$ with itself, and I strongly encourage you to drop it and adopt the standard one.

Some references:

1. Blyth, R. D.; Fumagalli, F.; Morigi, M. A survey of recent progress on non-abelian tensor squares of groups. Ischia group theory 2010, 26–38, World Sci. Publ., Hackensack, NJ, 2012. MR3184981

2. Russell D. Blyth and Robert Fitzgerald Morse. Computing the nonabelian tensor squares of polycyclic groups. J. Algebra 321 (2009), no. 8, 2139–2148. MR 2501513

3. Blyth, Russell D.; Fumagalli, Francesco; Morigi, Marta. Some structural results on the non-abelian tensor square of groups. J. Group Theory 13 (2010), no. 1, 83–94. MR2604847

4. R. Brown, D. L. Johnson, and E. F. Robertson, Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987), no. 1, 177–202. MR0913203

See also this previous answer.