CC BY
89
UDC 512.54
Siberian Journal of Science and Technology. 2018, Vol. 19, No. 2, P. 212-216
ON NECESSARY AND SUFFICIENT CONDITIONS OF SIMPLY REDUCIBILITY OF WREATH PRODUCT OF FINITE GROUPS
S. G. Kolesnikov
Reshetnev Siberian State University of Science and Technology 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation E-mail: [email protected]
A finite group is considered to be real if all the values of its complex irreducible characters lie in the field of real numbers. We note that the above reality condition is equivalent to the fact that each element of the group is conjugate to its inverse. A finite group is called simply reducible or a SR-group if it is real and all the coefficients of the decomposition of the tensor product of any two of its irreducible characters are zero or one.
The notion of a SR-group arose in the paper of R. Wiener in connection with the solution of eigenvalue problems in quantum theory. At present, there is a sufficient amount of literature on the theory of SR-groups and their applications in physics. The simplest examples of SR-groups are elementary Abelian 2-groups, dihedral groups, and generalized quaternion groups.
From the point of view of a group theory questions of interest are connected first of all with the structure of simply reducible groups. For example A. I. Kostrikin formulated the following question: how to express the belonging of a finite group to the class of SR-groups in terms of the structural properties of the group itself. Also, for a long time it was not known whether a simply reducible group is solvable (S. P. Stunkov's question). A positive answer to the last question was obtained in the works of L. S. Kazarin, V. V. Yanishevskiy, and E. I. Chankov. Questions concerning the portability of the properties of a group to subgroups, factor groups, and also their preservation in the transition to direct (Cartesian) and semidirect products or wreath products are always of interest.
The paper proves that the reality of H is the necessary condition of simply reducibility of the wreath product of the finite group H with the finite group K and the group K must be an elementary Abelian 2-group. We also indicate sufficient conditions for simply reducibility of a wreath product of a simply reducible group with a cyclic group of order 2.
Keywords: simply reducible group, real group, wreath product.
Сибирский журнал науки и технологий. 2018. Т. 19, № 2. С. 212-216
О НЕОБХОДИМЫХ И ДОСТАТОЧНЫХ УСЛОВИЯХ ПРОСТО ПРИВОДИМОСТИ
СПЛЕТЕНИЯ КОНЕЧНЫХ ГРУПП
С. Г. Колесников
Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31
E-mail: [email protected]
Конечную группу назовем вещественной, если все значения её комплексных неприводимых характеров лежат в поле вещественных чисел. Отметим, что сформулированное выше условие вещественности равносильно тому, что каждый элемент группы сопряжен со своим обратным. Конечная группа называется просто приводимой, или SR-группой, если она вещественна и все коэффициенты разложения тензорного произведения любых двух её неприводимых характеров равны нулю или единице.
Понятие SR-группы возникло в работе Р. Винера в связи с решением задач на собственные значения в квантовой теории. В настоящее время имеется достаточно литературы по теории SR-групп и их приложениям в физике. Простейшие примеры SR-групп дают элементарные абелевы 2-группы, диэдральные группы и обобщенные группы кватернионов.
С точки зрения теории групп прежде всего представляют интерес вопросы, связанные со строением просто приводимых групп. Например, А. И. Кострикин отмечает следующий вопрос: как выразить принадлежность конечной группы к классу SR-групп в терминах структурных свойств самой группы. Также продолжительное время не было известно, является ли просто приводимая группа разрешимой (вопрос С. П. Струнко-ва). Положительный ответ на последний вопрос был получен в работах Л. С. Казарина, В. В. Янишевского и Е. И. Чанкова. Вопросы, касающиеся переносимости свойств группы на подгруппы, фактор-группы, а также сохранения их при переходе к прямым (декартовым) и полупрямым произведениям или сплетениям, всегда вызывают интерес.
Доказано, что необходимым условием просто приводимости сплетения конечной группы Н с конечной группой К является вещественность Н, а группа К должна быть элементарной абелевой 2-группой. Также указаны достаточные условия просто приводимости сплетения просто приводимой группы с циклической группой порядка 2.
Ключевые слова: просто приводимая группа, вещественная группа, сплетение.
Doi: 10.31772/2587-6066-2018-19-2-212-216
Introduction. Simply reducible groups introduced by Wigner in [1] in connection with the questions of the quantum theory have found applications in other branches of physics as well (for example, [2-6]). The questions formulated above were noted in [7-9]. A positive solution of the problem of a finite simply reducible group solvability was published in a series of papers [10-12]. Questions of the portability of various properties of groups to wreath products were considered in [13-15].
In this paper we determine necessary and sufficient conditions for the simply reducibility of the wreath product of finite groups. In the first section we find necessary and sufficient conditions for the reality of the wreath product of two finite groups. In the second section we give necessary and sufficient conditions for the simply reducibility of the wreath product of a simply reducible group and a cyclic group of order 2, and we present an infinite series of simply reducible groups built using the construction of a wreath product.
The standard group-theoretical notation is used in this paper (for example, [16]). We also use the following equivalent definition of a simply reducible group that is more convenient for computations. A finite real group is called simply reducible if
X0G (g ) =ElQ (g )|2,
geG geG
where 6g(g)=|{h e G | h2 = g}| and Cg(g) is a centralizer of the element g .
Necessary conditions for simply reducibility of the wreath product of finite groups
Theorem 1. Let H and K be finite groups, and their wreath product HsK is a simply reducible group. Then H is a real group, K is an elementary Abelian 2-group.
Proof. According to its definition, a simply reducible group is real. We show that the reality G = HsK implies that H is real, and the group K must be an elementary Abelian 2-group.
So, let the group G = HsK be real, k1 = 1, k2,..., kn are all the different elements of the group K. Let us take an arbitrary element h e H and consider the element
x = hk1of the group G. According to the definition of the reality, there will be found such an element
y = kt • hf1...hknn, h1,...,hn e H,
of the group G, that xy = x_1. We have
(h-1)*1 = x— = xy = (h;')kn ...(h-1)*1 • k-1hk1 kt • hk1...h^ =
= (h-\)k1...(h;1hhi)ki...(h;\)kn = (h;lhht)ki,
whence it follows that kt =1 and h 1= h-lhht. Thus, every element of the group H is conjugate in it with its inverse and, hence, H is a real group.
Let us prove that K is an elementary Abelian 2-group. We assume the contrary, let | kj |= s >2 for some j. Without restricting the generality, we may assume that k2 = kt,k3 = k2,...,k„ = ks-1.Let us consider the element
2 j 3 j s J
g = k2 • hk of the group G, where 1 ^ h e H. Again according to the definition of the reality there will be found such an element
y = kt • hk ...hknn, h1,...,hn e H,
of the group G, that gy = g_1. We have
k2-1(h-1)k1k-1= g-1= gy =
= (h-l)kn ...(hr1/1 k-% hk1 kihik1 ...hknn =
k k = k2ki (h-1 )knk2 ... (hr1 )k1k2 hk1ki h1k1 . hknn ,
whence it follows that k^ = k-1 (in particular, kt does not lie in a cyclic subgroup (k2) ) and, therefore,
(h-')k1k-1 = (h;1)^-...(h1-1)k1k-1 hk1ki h1k1...hknn. (1)
Remembering, that k1k-1 = ks, k2k-1 = k1, k3k- = k2,..., kfi- = ks ^, we obtain (after comparing the right-hand and left-hand sides of relation (1)) the system of equations
1 = h-\,1 = h2-V = h-1h2,. ,1 = h-A_ 2, h = h-1, from which
h = (h;1h_1 )(h-1h,_2). (h-1h )(h-2lh )(h-lhs) = 1,
contradiction with the choice of h. Hence, the group K is an elementary Abelian 2-group. The theorem is proved.
We show that the conditions formulated in Theorem 1 are sufficient for the realness of the group G = HsK. Let G = HsK, where the group H is real, and K is an elementary Abelian 2-group. We choose an arbitrary element g e G and establish its realness.
Situation 1. Let g = h1k1... h^, where h1,..., hn e H. From realness H it follows that there exist such elements r,...,r e H, that hr' = h-1, i = 1,. n. The element
1 > ' n > I I > ''
y = rk1... r, obviously, inverts g.
Situation 2. Let g = k • h^1...hi", where k e K u h1,...,hn e H. Without restricting the generality, we may assume, that the elements of the group K are arranged in such a way that kik = k+m , i = 1,..., m = n / 2. According
to this the element g (and any other element from K) can be expressed by the following product
m
g = k-Uhk-hki.
1=1
We sort out for each product htht+m, i = 1,..., m, such an element f e H, that (hihi+m)f = (hihi+m )-1 and assume
m
y = k-n(f,h,)ki (Klfi)kk.
i=i
Then
m
gy = ^ (f )kik - kk x
i=1
mm
xn.h№mk-n(f.h, )ki (h-f )kik =
i=1 i=1
m
= k-U^f,-l)kik (f (fh, )ki (h;1fi )kik =
i=1
m
= k -U(fr'hihf )ki (h-f -1hh-1fi )¥ = i=1
m
= k-n(h-m )ki (Kl)k'k = g-1.
i=1
A sufficient condition of simply reducibility of the wreath product of a simply reducible group and a cyclic group of order 2
Theorem 2. In order that the wreath product G=HsZ2 of a simply reducible group H and a cyclic group of order 2 be a simply reducible group it is necessary and sufficient that the equality
X (3eH (u) | Ch (u) | +39H (u) | CH (u) |2 +1 CH (u) |3) =
(u,v)eH xH ,uevH
= 41H X Ch(h)|2 + 3 X I CH(h)|4. (2)
heH (h,f)eHxH ,hefH
be satisfied.
In the following three lemmas, the structure of the centralizers of the group G elements is determined.
Lemma 1. If g = (h, f), f, h e H, and fv = h for some v e H , then
Cg (g) = {(t,u), CT(uv,tv-1) 11 e Ch (h),u e Ch(f)},
in particular,
i ii i2 |CG(g)| = 2-\Ch(h)| .
Proof. Let z e CG (g). Let us cosider two situations. Situation 1. Let z = (x, y), x, y e H. Then
z-lgz = (x-1, y-1)(h, f)(x, y) = (hx, fy) = (h, f), from which x e CH (h) and y e CH (f).
Situation 2. Let z = ct(x, y), x, y e H. Then z-1= CT( y-1, x-1),
z"1 gz = a(y_1, x_1)(h, f )CT(x, y) = = (x-1, y-1 )(f, h)( x, y) = (fx, hy ) = (h, f),
from which fx = h and hy = f . From the equalities fx = h and fv = h it follows, that x = uv, u e CH (f). Similarly, from the equalities hy = f and hv = f it follows, that y = tv-1, where t e CH (h) . Thus, z = a(uv, tv-1), where t e CH (h) and u e CH (f).
The second assertion of the lemma follows from the isomorphism CH (h) = CH (f). The lemma is proved.
Lemma 2. If g = (h, f) and the elements h and f are not conjugate in H , then
Cg (g) = { (x, y) | x e Ch (h), y e Ch (f)}, in particular,
Cg (g)| = Ch (h)| - Ch (f )|. Proof. Let z e CG (g). If z = (x, y), then
gz = (x-1, y-1)(h, f)(x, y) = (hx, fy) = (h, f).
From which x e CH (h) and y e CH (f). If we assume that z = ct( x, y), then
gz = CT(y-\ x-1)(h, f )CT( x, y) = (fx, hy ) = (h, f).
which contradicts the disconjugacy of h and f. The lemma is proved.
Lemma 3. If g = a(h, f), then
Cg (g ) =
= {(yh, y)| y e Ch (h-f)} U{a(f "tth^1) 11 e Ch (f-h)}, in particular,
\Cg (g )| = 2 -\Ch (h-f )|.
Proof. Let z e CG (g). Again we consider two situations.
Situation 1. Let z = (x, y), x, y e H. We show that
z e CG (g) when and only when y e CH (hf) and x = yh. We have
CT(h, f) = z"1 gz = (x-1, y-')CT(h, f)(x, y) = = ct( y_1, x— )(h, f)(x, y) = ct( y _1hx, x fy), therefore
y-lhx = h, x-f = f. (3)
Multiplying these equalities, we obtain y~1hjy = hf , that is, y e CH (hf). The first equality (3) in this case gives x = yh.
Inversely, let y e CH (hf) and x = yh. Then and v = y2. If z = ct(x, y), then the equality z2= g is y-lhx = h and, consequently, x"1 fy = h- y~ lhfy = satisfied when and only when
= h lhf = f , that is, equalities (3) are satisfied, hence,
u = yx, v = xy. (6)
г ^gz = g
We show that the conditions (6) are equivalent to the fact that both u and v are conjugate in some element Situation 2. Let z = ct(x, y), x, y e H. Then h e h and
g = z-1gz = CT(y-',x-')a(h, f )ct(x,y)= x = hy = urh t e Ch(u)- (7)
= a(y-',x-')(f,h)(x,y) = ct(y"1 fx,x"'hy) Indeed, = ^, therefore, u and v are conjugate
in H. Let v = uh. Then from (6) it follows that y = ux-1
h —1 x
u = v = xy = xux = u
and the equality z 1gz = g is equivalent to the following and two equalities:
y~1 fx = h, x- lhy = f. (4) 1 1
From which x = h t, y = ut h for some t e CH (h).
Multiplying the first equality by the second from the The lemma is proved.
right, we obtain Using Lemma 4, we find the sum of the cubes of the
_1 values of the function 6. We have
y--fhy = hf = (fhf , £6G(g)= X 6G(a(u,v)) + X 6G((u,v)) =
geG u ,veH u,veH
from which y = h V for some t e CH( fh) and _3 „ n3 , _ N
H = X 0G((u,v)) = X 0H(u)0H(v)+
.. .,-tt . .,.,.,.,. h
x = f-1 yh = f -lth-lh = ff.
+ S (0H (u)0H (v)+ | Ch (u) |)3 = X 0H (u)0H (v) +
Inversely, let x = f ~lt and y = th- for some (u-v)eH xH-uevH u ,veH
t e CH ( fh). Then
+
I
(u ,v)eHxH ,uev'
(39H (u) | Ch (u)| + ^
v+ 39H (u) | Ch (u)|2 + | Ch (u)f j
(8)
y- f = ht- f -lt = h
and Comparing (5) with (8) and taking into account that,
x_ih = t-1 foth"1 = fhh= f by virtue of the simply reducibility of the group H , the
y , following equality is true
that is, equalities (4) are satisfied. The lemma is proved. X 63H (u)63H (v) = X |CH(h)|2 |CH(f )|2,
Using lemmas 1-3, we find the sum of the squares of u,veH h,f eh
the centralizers of the elements of the group G. We have ■ ■ , c
b r we obtain that condition (2) is necessary and sufficient tor
«x 12 v i ^ ^ 12 simply reducibility of the wreath product of a simply
\CG(g) = X | CG(a(h, f))|2 + X | CG((h, f))|2 = , i vi i- r j „ i,
geG GVO/| h/En G hfeH G reducible group with a cyclic group oi order 2. The
theorem is proved.
= X | 2CH(hf) |2 + X (2| CH(h)|2)2 + The proof of Theorem 2 allows us to construct the
,|2 + X (2| CH(h) |2)2 ■
hfiH ( hj f)eH xH hefH following infinite series of simply reducible 2-groups.
Corollary 1. Let G = HsZ2, where H is an elementary
+ X (|CH (h) 11 CH (f) |)2 = 41 H| X |CH (h)| + Abelian 2-group of order 2n. ThenG is simply reducible.
(h,f)EH,H,h f hEH Proof. Let Z2= {1,-1}, H = Z2 x...xZ2 and
+ X | CH(h) |2| CH(f)|2 +3 X | CH(h) |4. (5) h = (hl,hk). We find 6(h). ^v^us^ if
h,fEH f)eHxH■hEfH h = (1, . ,1), then 6(h) = |H = 2n . If h = -1 for at least
Lemma 4. Let g e G . Then one i, then 6(h) = 0 , as the equation x2 = -1 is not
1) 6G(g) = 0 , if g = ct(w, v); solvable in group Z2. As H is Abelian, then
2) 6g (g) = 6n (u)6h (v), if g = (u,v) and, u and v \cEn (h)\ = HI for any h e H . From which, are not conjugate in H; "
3) 6G(g) = 6H (u)6H (v) + |CH(u)|, if u and v are X conjugate in H. ("-v)EH xH '"Ev
Proof. The assertion 1) is obvious, since the square of = 24n (4 + 3 • 2n) =
any element from G lies in H x H . = , , ( ,2 3 ^ | C (,) |4
Let g = («,v) and z eG . If z = (x,y), then the = ^l»CH(h) + 3 X H1 Ch(h)| '
f 39H (u) | Ch (u)| + ^
v+ 39H(u) | Ch(u)|2 + | Ch(u)|3y
4n
equality z2 = g is satisfied when and only when u = x2 The corollary is proved.
Conclusion. It is proved that the reality of H is the necessary condition of simply reducibility of the wreath product of the finite group H with the finite group K, and the group K must be an elementary Abelian 2-group. We also indicate sufficient conditions for simply reducibility of a wreath product of a simply reducible group with a cyclic group of order 2.
Acknowledgments. The work was supported by the Russian Foundation for Basic Research, project No. 16-01-00707a.
Благодарности. Работа поддержана Российским фондом фундаментальных исследований, проект № 16-01-00707а.
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© Kolesnikov S. G., 2018
