Abel's theorem in problems and solutions - School of Mathematics
Abel's theorem in problems and solutions - School of Mathematics
Abel's theorem in problems and solutions - School of Mathematics
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146 Problems <strong>of</strong> Chapter 1<br />
permutation <strong>and</strong> together with it the permutation<br />
Suppose now that the normal subgroup N conta<strong>in</strong>s a permutation<br />
<strong>of</strong> the type (b) (see 190) <strong>and</strong> that is an arbitrary<br />
permutation. Choose the elements <strong>and</strong> different from <strong>in</strong><br />
the set Either <strong>in</strong> the row or <strong>in</strong> the row<br />
there will be an even number <strong>of</strong> <strong>in</strong>versions. Consequently<br />
one <strong>of</strong> the permutations or<br />
is even. Denot<strong>in</strong>g this permutation by we obta<strong>in</strong> that <strong>in</strong> all the cases<br />
N conta<strong>in</strong>s the permutation<br />
If the normal subgroup conta<strong>in</strong>s a permutation <strong>of</strong> the type (c) (see<br />
190)‚ for example‚ the permutation then it also conta<strong>in</strong>s<br />
all permutations <strong>of</strong> the type (c). Indeed‚ <strong>in</strong> this case one <strong>of</strong><br />
the permutations or is even.<br />
Let this permutation be denoted by Thus N conta<strong>in</strong>s the permutation<br />
192. We will count the number <strong>of</strong> permutations <strong>of</strong> each one <strong>of</strong> types<br />
(a)‚ (b) <strong>and</strong> (c) (see 190).<br />
a) There exist 5 · 4 · 3 · 2 · 1 = 120 sequences <strong>of</strong> 5 numbers 1‚ 2‚ 3‚ 4‚ 5.<br />
S<strong>in</strong>ce every permutation <strong>of</strong> the type (a) can be written <strong>in</strong> five equivalent<br />
ways (depend<strong>in</strong>g upon the choice <strong>of</strong> the first element) the number <strong>of</strong><br />
permutations <strong>of</strong> the type (a) is 120/5 = 24.<br />
b) By the same reason<strong>in</strong>g as <strong>in</strong> the case (a) one obta<strong>in</strong>s that the<br />
number <strong>of</strong> permutations <strong>of</strong> the type (b) is equal to 5 · 4 · 3/3 = 20.<br />
c) A permutation <strong>of</strong> the type (c) can be written <strong>in</strong> 8 different ways<br />
(one can choose <strong>in</strong> 4 ways <strong>and</strong> afterwards <strong>in</strong> two ways). Hence the<br />
number <strong>of</strong> permutations <strong>of</strong> type is equal to 5 · 4 · 3 · 2/8 = 15.<br />
Every normal subgroup N conta<strong>in</strong>s the unit element. Moreover‚ from<br />
the result <strong>of</strong> Problem 191 it follows that a normal subgroup <strong>of</strong> the group<br />
either conta<strong>in</strong>s all the permutations <strong>of</strong> a given type (see 190) or it<br />
conta<strong>in</strong>s none <strong>of</strong> them. The order <strong>of</strong> a normal subgroup must divide the<br />
order <strong>of</strong> the group (60). But add<strong>in</strong>g to the number 1 the numbers<br />
24‚ 20‚ <strong>and</strong> 15 one obta<strong>in</strong>s a divisor <strong>of</strong> 60 only <strong>in</strong> two cases: when one<br />
adds noth<strong>in</strong>g <strong>and</strong> when one adds all the three numbers. The first case<br />
corresponds to the unit subgroup‚ the second one to the whole group<br />
193. Such a subgroup is‚ for example‚ the subgroup conta<strong>in</strong><strong>in</strong>g all