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A note on normal and nonnormal subgroups of given index
INFORMATION SCIENCES 22, 245-246 (1980)
245
A Note on Normal and Nonnormal Subgroups of Given Index
F. B. CANNONITO Department of Mathematics, Unioersity of California, lroine, Cah'.fornia92717
A well-known elementary proposition of group theory of interest in algebraic coding theory asserts that any subgroup of index 2 must be normal. Less well known is its generalization: ifp is the smallest prime dividing the order of the group, any subgroup of index p is normal. In this note we give an elementary proof of the generalization and, by way of contrast, show for each integer n > 2 how to consm~ct a group Gn with nonnormal subgroup lln of index n. First, the generalization. PROPOSITION 1. Let G be a finite group of order n, and let p be the smallest prime divisor of n. Then if H is a subgroup of index p in G, H is normal.
Proof. First, for any x ~ H consider H, x H , . . . , x e - I H . These cosets are distinct and exhaust G, for if not, there is a first k such that xkHffixIH
(l
But then x k - I H = H , whence 1--0 and k , p - I. Let m be the order of x. Then mffigk+r, r < k , and thus l ffixm=(xk)qx ". Since x k E H , we have x'HffiH, and this contradicts the choice of k unless r = 0 . It follows (Xk)~=.(Xq) k=, 1 and so kin. But k ~ p - 1 , and so again we have a contradiction, as any prime divisor of k must necessarily be smaller than p and divide n. We are now able to finish the proof. Suppose, for some x E G and h e l l , that y - - - x h x - I ~ l l . Then H, y H , . . . , y P - I l l are distinct and exhaust G. It follows xffiy'h I for some r < p - I and hi in H. Then x f x h ' x - l h t and so h'x-thlffil or, equivalently, x f h l h " , an element of H. This contradiction concludes the proof. W e n o w show
PROPOSITION 2. For each integer n > 2 there is a group G~ with nonnormal subgroup H~ of index n in G~.
Proof. Let p be an odd prime. Put De as the group with generators a, b subject to a 2ffiffi1, b e-- 1, and (ab) 2-- 1. Evidently De is the dihedral group of order 2p. Clearly, De is nonabelian for a b a - I - - b -~, and b ~ b -I. By elementary counting arguments or Sylow theory De has a subgroup Hp of order 2 and ©Elsevier North Holland, Inc., 1980 52 Vand~bilt Ave., New York, NY 10017
00204~5/80/08245-02501.75
246
F.B. CANNONITO
index p. This subgroup is not normal lest Dp be abelian, for then by Sylow theory Dp is the direct product of the cyclic groups Hp and DJHp. Now, let n =pT,. - - PT" be the prime decomposition of n > 2. For each prime Pi (1 < i < r) find Dp, and Hp, as above. Then we can take the nonabelian group at
a2
a , = ( D , , x . . . xD,,) x (D, x . . . xD,=) x . - and ~I
n . = (/4,, x . . - x A r , ) x
a2
(/4, x - - - x n , 2 ) x . . - .
First, note that H. can not be normal in G., for then G. would be abelian as the direct product of its abelian subgroups H. and (3./1t.. It follows that G. and H. satisfy the conditions of the proposition, and the proof is complete.
Received June 1980
245
A Note on Normal and Nonnormal Subgroups of Given Index
F. B. CANNONITO Department of Mathematics, Unioersity of California, lroine, Cah'.fornia92717
A well-known elementary proposition of group theory of interest in algebraic coding theory asserts that any subgroup of index 2 must be normal. Less well known is its generalization: ifp is the smallest prime dividing the order of the group, any subgroup of index p is normal. In this note we give an elementary proof of the generalization and, by way of contrast, show for each integer n > 2 how to consm~ct a group Gn with nonnormal subgroup lln of index n. First, the generalization. PROPOSITION 1. Let G be a finite group of order n, and let p be the smallest prime divisor of n. Then if H is a subgroup of index p in G, H is normal.
Proof. First, for any x ~ H consider H, x H , . . . , x e - I H . These cosets are distinct and exhaust G, for if not, there is a first k such that xkHffixIH
(l
But then x k - I H = H , whence 1--0 and k , p - I. Let m be the order of x. Then mffigk+r, r < k , and thus l ffixm=(xk)qx ". Since x k E H , we have x'HffiH, and this contradicts the choice of k unless r = 0 . It follows (Xk)~=.(Xq) k=, 1 and so kin. But k ~ p - 1 , and so again we have a contradiction, as any prime divisor of k must necessarily be smaller than p and divide n. We are now able to finish the proof. Suppose, for some x E G and h e l l , that y - - - x h x - I ~ l l . Then H, y H , . . . , y P - I l l are distinct and exhaust G. It follows xffiy'h I for some r < p - I and hi in H. Then x f x h ' x - l h t and so h'x-thlffil or, equivalently, x f h l h " , an element of H. This contradiction concludes the proof. W e n o w show
PROPOSITION 2. For each integer n > 2 there is a group G~ with nonnormal subgroup H~ of index n in G~.
Proof. Let p be an odd prime. Put De as the group with generators a, b subject to a 2ffiffi1, b e-- 1, and (ab) 2-- 1. Evidently De is the dihedral group of order 2p. Clearly, De is nonabelian for a b a - I - - b -~, and b ~ b -I. By elementary counting arguments or Sylow theory De has a subgroup Hp of order 2 and ©Elsevier North Holland, Inc., 1980 52 Vand~bilt Ave., New York, NY 10017
00204~5/80/08245-02501.75
246
F.B. CANNONITO
index p. This subgroup is not normal lest Dp be abelian, for then by Sylow theory Dp is the direct product of the cyclic groups Hp and DJHp. Now, let n =pT,. - - PT" be the prime decomposition of n > 2. For each prime Pi (1 < i < r) find Dp, and Hp, as above. Then we can take the nonabelian group at
a2
a , = ( D , , x . . . xD,,) x (D, x . . . xD,=) x . - and ~I
n . = (/4,, x . . - x A r , ) x
a2
(/4, x - - - x n , 2 ) x . . - .
First, note that H. can not be normal in G., for then G. would be abelian as the direct product of its abelian subgroups H. and (3./1t.. It follows that G. and H. satisfy the conditions of the proposition, and the proof is complete.
Received June 1980