# The finite simple groups

## Robert A. Wilson

Reviews | Ordering information | Errata | Extra material

## Information

This book was published on 23rd December 2009, as vol. 251 in the Springer Graduate texts in mathematics' series. Details here.

## Reviews

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. Half-way 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 3-cycles in H centralize $$A_4$$ that is important, not that the 3-cycles 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 $$C_{(n, q-1)}:(C_2×C_d)$$ and presentation $$\langle x, y, z\mid x^{(n, q-1)}= y^2= z^d= x^yx=1, x^z=x^p, [y, z]=1\rangle$$. This is the same as originally stated only in the case when $$(n, q-1)=(n, p-1)$$.
• 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 non-zero 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, 85-6: 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 $$(n-2k)$$ instead of $$(n-k)$$.
• 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 right-multiplication'
• 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 $$(B(1-i_0))(1+i_t) = (BL)R=LR=2B$$.
• 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 $$\mathbf P \Omega_8^+(3){:}S_4$$

## Extra material

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.