**INTRODUCTION**

The problem originated from the study of alternating group A_{7}
on 7 letters, by Liggonah (1977). It is the generalization of the proposition
in the proof of the main result established by Liggonah (1977).

The notations used are standard, as used and defined in Gorenstein (1968).
Conditions (i) through (iii) are conditions on the subgroup H of G, Giving
the structure of H. These are the normalizer conditions referred to in
the theorem.

The proof of the theorem is by contradiction. We suppose that G is simple
and satisfies the conditions of the theorem and aim at arriving at a contradiction.The
method used in the proof involves studying the fusion of involutions in
H, By using character and group theoretic methods.

We divide the proof through a series of lemmas. From now on, G is assumed
to be simple and satisfies conditions (i) through (iii).

**
**PROOF OF THE THEOREM

**Lemma 1 **

For each class L of involutions of H, either there is an element l ε
L such that l^{-1}yl = y^{-1} or L⊂O_{3}`(H).

**Proof **

Conditions (i) to (iii) imply that N_{G}(X) = H = N_{G}(KxT).
Since KxT admits <y><t> and y acts fixed-point-free, KxT is
unique in (KxT)<t> and of class ≤ 2 by Stephen and Tyrer (1973a,
b). Hence (KxT)<t> is a Sylow 2-subgroup of its own normalizer,
so it is a Sylow 2-subgroup of G. Consider H_{1}=(RxKxT)<y><t>.
The principal 3-block B_{0}(H_{1}) of H_{1} has
the form:

e |
y |
L |

1 |
1 |
1 |

d |
1 |
δ |

d+1 |
-1 |
δ+1 |

If l^{-1}yl = y^{-1}is false for all l εL, then #; (l`l`
= sy) = 0 for all 3`-elements s commuting with y, by Higman (1968); where #(l`l`
= sy) is the number of conjugates of l with product equal to sy. Applying Higman
(1968) results, we

#(l`l` = sy) = |H_{1}|/|CH_{1}(y)|Σ
χ (*l*)^{2} χ(y) = 0 |

where, summation is over all characters χ in B_{0}(H_{1}).
Then this implies that

Σ χ(y)χ(*l*)^{2} = 0 giving
1+δ^{2}/d - (δ+1)^{2}/(d+1) = 0. |

That is, (δ–d)^{2} = 0, giving δ = d. Then l lies in
the kernel of every character in the principal 3-block of H_{1}. By
Brauer (1964a, b), l lies in O_{3}`(H_{1}) = O_{3}`(H)
and O_{3}`(H) = RxKxT and hence L⊂O_{3}`(H) as required.

Hence we have that a class L of involutions in H, either there is an
element lεL such that l^{-1}yl = y^{-1} or L⊂O_{3}`(H).

**Corollary
**

The involutions of H not in X = RxKxT are all conjugate (even in H).

**Proof**

The extended centralizer of y in H_{1} is C*H_{1}(y) =
<y><t>, so all involutions of H_{1} not in X must
be in <y><t>
D_{6} by Lemma 1 (otherwise they will lie in O_{3}`(H_{1})
= X). Using the fact that all involutions of D_{6} are conjugate
and C*_{H}(y) = P<t>, all involutions of H not in O_{3}`(H)
= O_{3}`(H_{1}) = X are conjugate to t as required.

We have seen in the proof of Lemma 1 that KxT is weakly closed in (KxT)<t>,
a Sylow 2-subgroup of G and of class 2 and any involution of H is conjugate
in H to t or lies in KxT. Since Ω_{1}(Z(KxT)) is characteristic
in KxT, elements of Ω_{1}(Z(KxT)) are conjugate in G only
if they are conjugate in N_{G}(Ω_{1}(Z(KxT))).

By maximality of H and the supposition that G is simple, we must have
that N_{G}(Ω_{1}(Z(KxT))) = H. In particular, if k
is an element of Ω_{1}(Z(KxT)), its conjugates in Ω_{1}(Z(KxT))
are k, k^{y}, k^{y2}. Since |Ω_{1}(Z(KxT))|>4,
we can always pick k so as not to be conjugate to t in H. From now on,
we assume that such a k has been picked.

**Lemma 2 **

The only conjugates of k in G lying in (KT)<t> are k, k^{y},
k^{y2}.

**Proof **

By the corollary and the underlying assumption, any further conjugates
k^{g} lies in KxT, whence k^{g}, k, k^{y}, k^{y2}
generate an abelian group. Let A be a subgroup of KxT chosen such that:

• |
A contains the greatest possible number of conjugates
of k |

• |
A is as large as possiblee as possible |

We first show that the conjugates of k lying in A are already conjugate in
N_{G}(A). Indeed, let kA.
Then k^{g}A^{g}
and so≤
C_{G}(k). Since (KxT)<t> is a Sylow 2-subgroup of G and of C_{G}(k),
we can assume ≤
(KxT)<t>. An element of (KxT)<t> not in KxT transforms k^{y}
to k^{y2} since it must involve t, so that an abelian subgroup of (KxT)<t>
not in KxT can not contain k^{y} or k^{y2}.

Furthermore, the conjugates of k in lie
in KxT. Thus, if is
not contained in KxT, then (
(KxT)) <k^{y}> is an abelian subgroup of KxT containing more conjugates
of k than A, contrary to the choice of A. Hence ≤
KxT.

By choice of A, each A and
are maximal abelian subgroups of KxT and since KxT is of class at most 2, (KxT)`
= Z(KxT) = A and because [KxT, A] = (KxT)` = A and applying theorem 2.1 in Gorenstein
(1968), page 18, gives A,
are normal in KxT. By weak closure of KxT in (KxT)<t>, it means that A
and
are conjugate in N_{G}(KxT) which is H. Since y acts fixed-point-free,
<A,>
is abelian by Stephen and Taylor, so it implies A =
by maximality of A. Thus the conjugates of A in H are A and A^{t}. Replacing
g by tg if necessary, we can assume A = A^{g}, which implies g εN_{G}(A).
Hence conjugates of k lying in A are already conjugate in N_{G}(A).
A is normalized by P so N_{G}(A) = XP and hence N_{G}(A) = H
or N_{G}(A) = XP. This implies the only conjugates of k in A are k,
k^{y}, k^{y2} as required.

**Lemma 3**

The element t is not conjugate to any element of KxT in C_{G}(k).

**Proof **

By Lemma 2, the conjugates of k are k, k^{y}, k^{y2}
and kk^{y}k^{y2} = 1 because y fixes kk^{y}k^{y2}
and y acts fixed-point-free. This implies the only conjugates of k^{y}<k>
in C_{G}(k)/<k> is k^{y}<k>. By Glaubermann
(1966), it implies that k^{y}<k> ε Z(C_{G}(k)/<k>).
This means that for some normal subgroup M of C_{G}(k) of odd
order, M<k, k^{y}> is normal in C_{G}(k). By Frattini
argument, this gives C_{G}(k) = M N_{CG(k)}(<k, k^{y}>).
Since <k, k^{y}> ≤ Ω_{1}(Z(KxT)) and because
of the fact that the conjugates of k in H lying in (KxT)<t> are
k, k^{y}, k^{y2}, then <k, k^{y}> = H. Hence
by maximality of H, we have N_{G}(<k, k^{y}>) =
H. Hence C_{G}(k) = MX<t> or C_{G}(k) = MX<x><t>.
Since KxT ≤ X, we have t is not conjugate to any element of KxT from
the structure of C_{G}(k) as required.

We could stop here, at this stage, by quoting Goldschmidt`s results,
presented by Thompson (1968), concerning strongly closed abelian 2-subgroups
because we have shown that <k, k^{y}> is a strongly closed
abelian 2-subgroup of C_{G}(k). But we can also finish more concisely.

**Lemma 4**

The element t is conjugate in C_{G}(k) to some element of KxT.

**Proof**

By Thompson`s Transfer Theorem given by Thompson (1968), t is conjugate in G
to some element of KxT, say t^{g}KxT.
then both k^{g} and <k, k^{y}> centralize t^{g},
so we can chose h in C_{G}(t^{g}) so that k^{gh} lies
in the same Sylow 2-subgroup of C_{G}(t^{g}) as <k, k^{y}>.
By Lemma 3, a Sylow 2-subgroup of G contains only three conjugates of k, so
this implies k^{gh} = k^{y2} for some i = 1,2,3. Since hC_{G}(t^{g}),
t^{ghy-1} lies in KxT and ghy^{-i} C_{G}(k),
so t is conjugate in C_{G}(k) to some element of KxT, proving the lemma.

The contradiction between Lemma 3 and Lemma 4 completes the proof of
the theorem. That is, G is not simple.