
The finite simple groups


Robert A. Wilson 
Information
This book was published on 23rd December 2009, as vol. 251 in the
Springer `Graduate texts in mathematics' series.
Details here.
`The book under review has as its main goal to give an introductory overview of the construction and main properties of all finite simple groups. ... This book is the first one that attempts to give a systematic treatment of all finite simple groups, using the more recent and efficient constructions. ... The author succeeds in making this important but difficult area of mathematics readily accessible to a large sector of the mathematical community, and for this we should be grateful.' (Felipe Zaldivar, The Mathematical Association of America, March, 2010)
 The full review is here
`This book is a unique introductory overview of all the finite simple groups, and thus it is suitable not only for specialists who are interested in finite simple groups but also for advanced undergraduate and graduate students in algebra.'
 from the review by Hiromichi Yamada in Zentralblatt für Mathematik.
`In summary, the book brings much information to the classroom. It contains exactly those things one would like to know if one were to meet the individual simple groups for the first time. ... The way the book is written makes things accessible also to those who are not great experts in group theory. Anyone interested in finite groups, in particular in finite simple groups, should have this book on his or her bookshelf.'
 from the review by Gernot Stroth in Mathematical Reviews.

A number of errors have arisen from a careless
(and inconsistent) definition of the real part of
an octonion. Halfway down p. 123, the definition
of `real part' is inconsistent with p.121 l.5. What is
here called the `real part' should be called the `trace'
that is \(\mathrm{Tr}(\sum_i\lambda_ix_i)=\lambda_4+\lambda_5\),
since it is actually twice the real part when the
characteristic is not 2.
As a consequence, replace Re by Tr in the following places:
p.132 l.16,
p.145 throughout,
p.151 l.10
Also replace `real parts' by `traces' on p.132 l.18.
This is compounded by a separate error on p.150 l.4,
where Re(DEF+DFE)/2 should be Re(DEF).
Consequently, replace Re(DEF+DFE), or equivalent, by Tr(DEF) in:
p.152 l.10,
p.168 l.5
p.181 l.2
 p.VIII, l.1: the quotation should be attributed to Richard Hamming,
not Gauss. The italics have been added.
 p.18, l.23: it is the fact that the 3cycles in H centralize \(A_4\) that is important,
not that the 3cycles in \(A_8\) centralize \(A_5\).
 p.28, l.2: there is a minus sign missing in the second line of (2.12).
 p. 50.
Theorem 3.2 (i) is incorrectly stated. The group has shape \(C_{(n, q1)}:(C_2×C_d)\) and presentation \(\langle x, y, z\mid x^{(n, q1)}= y^2= z^d= x^yx=1, x^z=x^p, [y, z]=1\rangle\). This is the same as originally stated only in the case when \((n, q1)=(n, p1)\).
 p.63, para. 5: in the case \(k=1\), in characteristic 2, the group Q
is elementary abelian, not special.
 p.65, l.6: \(z_n\) should be the number of nonzero vectors of norm 0.
 p.67, l.3ff: this holds only if \(k\ne n/2\). If \(k=n/2\), then Q is elementary abelian.
 p.82, 856: in the statement of Theorem 3.5, one case has inadvertently been omitted.
The same applies to the summary in the first paragraph of p.82. The omitted case is:
(vii) A semilinear group \(\Gamma\mathrm{L}_1(q^n)\).
The other semilinear groups do not need to be mentioned, since they are almost simple
modulo scalars. The sketched proof needs more care, as the field is not
algebraically closed, so it needs consideration of the case when the representation
is irreducible over the ground field but not over an extension field.
 p.93: in Theorem 3.9, the stabilizer of an isotropic \(k\)space should have \((n2k)\) instead of \((nk)\).
 p.107: in Exercise 3.17 one needs the hypothesis (which is assumed throughout but which should perhaps be explicitly mentioned from time to time) that \(f(u,v)=0\) if and only if \(f(v,u)=0\).
 p.111, l.1: \(n=1\) should read \(n=0\).
 p.144, Theorem 4.3(vi): needs the extra condition \(r\ne 3\).
 p.149, (4.80) is wrong. The correct inner product is \(\frac12\mathrm{Tr}(x\overline{y}^\top+y\overline{x}^\top)=\mathrm{Tr}(x\circ y)\).
 p.166, Theorem 4.5(ix): the subgroup \((q\sqrt{2q}+1)^2{:}4S_4\) is maximal only if \(q>8\).
 p.188, l.4: it is perhaps not as easy as I claimed to see that the sextet has all
of the stated properties. It is clear that the sum of the original tetrad and any of
the other five is an octad, but it is not so clear that the sum of two of the last five
tetrads is also an octad. To see this, put the original tetrad in the first column of the MOG,
and the other tetrads in the other columns, and
note first that
since any two octads intersect evenly we have the parity
condition: every octad either intersects all columns evenly,
or intersects all columns oddly.
Now if the sum of any two columns is not an octad, then the octad P containing
one column and one point of the other column has distribution \((4,2,2,0,0,0)\) across
the columns. Consider the octad Q containing three points of the first column,
and two points of the second column, one of which is in P and one of which is not.
Then Q cannot intersect the columns evenly, for then it would intersect P in
at least five points; and it cannot intersect the columns oddly, for then it would
have more than eight points. This contradiction proves the claim.
(This argument is a modified version of Lemma 1 from R.T.Curtis's PhD thesis.)
 p.189, l. 12: \((3,1^5)\) should read \((3,1)\)
 p.197, l.15: \(3^3\) should read \(3^2\)
 p.197, l.2: the justification `since two hexads cannot intersect in exactly
one point' can be deleted, since the 1 in the bottom left corner of the Leech triangle
says that there exist two hexads with empty intersection.
 p.227, l.1/2: `conjugation' should be `rightmultiplication'
 p.232, the second equation of (5.79) is wrong and should be
deleted. The second line of (5.80) should be replaced by the
simpler (and less incorrect) argument
\((B(1i_0))(1+i_t) = (BL)R=LR=2B\).
 p.233, l. 4: there should be a factor \(\overline{s}\) after the last )
 p.261, Table 5.7: there is a group missing from the list, namely
\(\mathbf P \Omega_8^+(3){:}S_4\)
Here I will add some links to extra material that did not make it into the book.
Last updated 5th March 2017.
This page is maintained by Robert Wilson. The
views and opinions expressed in these pages are mine. The contents of these
pages have not been reviewed or approved by Queen Mary, University of
London.