1
00:00:01,610 --> 00:00:04,370
PROFESSOR: Two particles is
interesting in many cases.

2
00:00:04,370 --> 00:00:07,880
But in order to see
what really is happening

3
00:00:07,880 --> 00:00:11,150
and how much structure you
have to go to more than two

4
00:00:11,150 --> 00:00:12,270
particles.

5
00:00:12,270 --> 00:00:16,520
Two particles is a
little too special.

6
00:00:16,520 --> 00:00:19,040
So we need to go
beyond two particles

7
00:00:19,040 --> 00:00:22,110
and see what happens
with this operator.

8
00:00:22,110 --> 00:00:26,420
So I'll do that.

9
00:00:26,420 --> 00:00:28,880
So we'll add more particles.

10
00:00:31,800 --> 00:00:35,120
So let's do that.

11
00:00:35,120 --> 00:00:45,730
So if you have n particles,
n particles, capital n,

12
00:00:45,730 --> 00:00:48,370
we can consider
also permutations.

13
00:00:48,370 --> 00:00:51,850
And how many permutations
are you going to have?

14
00:00:51,850 --> 00:00:56,090
You're going to have n
factorial permutations.

15
00:01:00,030 --> 00:01:04,890
And notation of Hilbert
space, n factorial permutation

16
00:01:04,890 --> 00:01:06,690
operators.

17
00:01:06,690 --> 00:01:09,540
So why you say n factorial?

18
00:01:09,540 --> 00:01:14,340
Because a permutation is a
reordering of your n objects.

19
00:01:14,340 --> 00:01:17,730
And therefore, in order
to define the reordering,

20
00:01:17,730 --> 00:01:22,330
you must decide which object
is going to be the new first.

21
00:01:22,330 --> 00:01:24,230
You have your n objects there.

22
00:01:24,230 --> 00:01:28,680
So you have n choices to pick
a first and n minus 1 choices

23
00:01:28,680 --> 00:01:30,930
to be a second, n minus 2.

24
00:01:30,930 --> 00:01:36,740
So n factorial is the
number of permutations.

25
00:01:36,740 --> 00:01:40,700
The permutations form a group.

26
00:01:40,700 --> 00:01:54,020
So the permutations
of n objects form

27
00:01:54,020 --> 00:02:09,550
this symmetric group s capital
n with n factorial elements.

28
00:02:14,060 --> 00:02:20,060
Just like you study
the group su2 of two

29
00:02:20,060 --> 00:02:26,180
by two unitary matrices,
or the group so3 of three

30
00:02:26,180 --> 00:02:29,330
dimensional rotations in space.

31
00:02:29,330 --> 00:02:31,520
The permutation
group is the group

32
00:02:31,520 --> 00:02:33,560
that deserves a lot of study.

33
00:02:33,560 --> 00:02:36,440
It's a finite discrete group.

34
00:02:36,440 --> 00:02:40,040
It's not a continuous
group of transformations.

35
00:02:40,040 --> 00:02:42,500
It's discrete transformations.

36
00:02:42,500 --> 00:02:45,750
And they do funny things.

37
00:02:45,750 --> 00:02:47,840
So we need to
understand this group so

38
00:02:47,840 --> 00:02:52,430
let's describe the notation
of what is a permutation.

39
00:02:52,430 --> 00:02:58,550
Suppose you have n
equal 4, four particles.

40
00:02:58,550 --> 00:03:06,590
So a given state is
represented by a, b, c, d.

41
00:03:06,590 --> 00:03:09,770
And you could say this is the
first particle, second, third,

42
00:03:09,770 --> 00:03:10,790
and fourth.

43
00:03:10,790 --> 00:03:12,890
And these are the states.

44
00:03:12,890 --> 00:03:15,020
And now we're going to
write the perturbation.

45
00:03:15,020 --> 00:03:21,770
The best thing, I think, is
just do an example, p3142.

46
00:03:21,770 --> 00:03:29,400
So a permutation now has capital
n labels, so four labels.

47
00:03:29,400 --> 00:03:34,320
And this is a permutation
acting on this state.

48
00:03:34,320 --> 00:03:38,150
Remember the 2-1
permutation exchange.

49
00:03:38,150 --> 00:03:40,620
The state in 2 was put in 1.

50
00:03:40,620 --> 00:03:42,780
The state in 1 was put in 2.

51
00:03:42,780 --> 00:03:45,810
Here is the instruction.

52
00:03:45,810 --> 00:03:52,920
So the instruction is, put in
position one the third state.

53
00:03:52,920 --> 00:03:55,800
So here it says 3,
so the third state

54
00:03:55,800 --> 00:03:58,570
goes into the first position.

55
00:03:58,570 --> 00:04:03,350
So that's the first position.

56
00:04:07,180 --> 00:04:12,970
The first state goes
into the second position.

57
00:04:12,970 --> 00:04:16,490
The fourth state goes
into the third position.

58
00:04:16,490 --> 00:04:17,620
The third in this here.

59
00:04:17,620 --> 00:04:20,320
So the third position
is now occupied

60
00:04:20,320 --> 00:04:24,160
by the fourth state, d.

61
00:04:24,160 --> 00:04:31,390
And the second state goes
into the fourth position, b.

62
00:04:31,390 --> 00:04:37,420
So this is the way we are
going to define permutations.

63
00:04:37,420 --> 00:04:39,430
People define them
in different ways

64
00:04:39,430 --> 00:04:43,360
and you have to be sure
you know what it means.

65
00:04:43,360 --> 00:04:45,060
But I'll right this.

66
00:04:45,060 --> 00:04:56,695
This means put third
object in first position.

67
00:04:59,200 --> 00:05:07,660
Here the second one, let's
write put first object

68
00:05:07,660 --> 00:05:09,055
in second position.

69
00:05:24,100 --> 00:05:30,910
It's nice to have this notation
because now you can understand

70
00:05:30,910 --> 00:05:33,700
how the group works.

71
00:05:33,700 --> 00:05:39,760
A group of
transformations is a set

72
00:05:39,760 --> 00:05:44,560
of elements that
can be multiplied.

73
00:05:44,560 --> 00:05:47,320
There is an identity.

74
00:05:47,320 --> 00:05:51,730
And things have
inverses as well.

75
00:05:51,730 --> 00:05:55,360
So permutations have inverses.

76
00:05:55,360 --> 00:05:58,120
If you rearrange the
objects in one way,

77
00:05:58,120 --> 00:06:02,150
you can rearrange them back.

78
00:06:02,150 --> 00:06:06,640
So for example, what
would be the inverse

79
00:06:06,640 --> 00:06:09,040
of this permutation?

80
00:06:09,040 --> 00:06:20,110
I will try to figure out
what is the inverse of p3142.

81
00:06:20,110 --> 00:06:23,650
So I look at this
thing and I think

82
00:06:23,650 --> 00:06:28,540
of what is the instruction
that I must put in order

83
00:06:28,540 --> 00:06:31,090
to rearrange back this series.

84
00:06:31,090 --> 00:06:36,520
And I say, well, I must put the
second back in position one.

85
00:06:36,520 --> 00:06:41,290
So there should be a p2 here.

86
00:06:41,290 --> 00:06:45,250
I should put the
fourth in position two.

87
00:06:50,430 --> 00:06:55,485
And then I should put the
first in position three.

88
00:07:00,090 --> 00:07:03,940
First in position three.

89
00:07:03,940 --> 00:07:08,185
And I should put the
third in position four.

90
00:07:11,210 --> 00:07:17,060
So I claim that is the
inverse of this operator,

91
00:07:17,060 --> 00:07:22,280
p213 is the inverse of this.

92
00:07:22,280 --> 00:07:25,670
So you can check it.

93
00:07:25,670 --> 00:07:42,920
Again, p2413 times p3142,
you just do it on a, b, c, d.

94
00:07:42,920 --> 00:07:50,870
Well, the first one, you
already know, is 2413,

95
00:07:50,870 --> 00:07:54,350
is put the third in one.

96
00:07:54,350 --> 00:07:57,710
Put the first in two.

97
00:07:57,710 --> 00:08:01,490
Put the fourth in three.

98
00:08:01,490 --> 00:08:06,380
And put the second in fourth, b.

99
00:08:06,380 --> 00:08:09,770
And then it says, put
the second first--

100
00:08:09,770 --> 00:08:11,690
so that's an a.

101
00:08:11,690 --> 00:08:15,860
Put the fourth, second, b.

102
00:08:15,860 --> 00:08:20,000
Put the first, third, c.

103
00:08:20,000 --> 00:08:24,860
And put the third, fourth, d.

104
00:08:24,860 --> 00:08:27,420
So it's back to
the original one.

105
00:08:27,420 --> 00:08:30,340
So indeed, it is the inverse.

106
00:08:30,340 --> 00:08:34,400
So you can write your
permutations and play with them

107
00:08:34,400 --> 00:08:38,510
and figure out how to
multiply any two permutations.

108
00:08:41,690 --> 00:08:45,110
More generally, we'll write
the permutation as follows.

109
00:08:47,790 --> 00:08:56,930
Suppose you have more generally
a permutations [INAUDIBLE]

110
00:08:56,930 --> 00:09:01,990
with some index alpha,
collective index alpha,

111
00:09:01,990 --> 00:09:07,360
that says how the
integers, 1 up to n,

112
00:09:07,360 --> 00:09:16,870
are rearranged into alpha of 1,
alpha of 2, up to alpha of n.

113
00:09:19,710 --> 00:09:24,480
So think of a permutation
and thinking n integers

114
00:09:24,480 --> 00:09:26,460
and rearranging them.

115
00:09:26,460 --> 00:09:32,550
So the first one,
the new first index

116
00:09:32,550 --> 00:09:35,130
is going to be alpha
of 1, alpha of 2.

117
00:09:35,130 --> 00:09:42,960
And alpha is a rule that takes
the n integers from 1 to n

118
00:09:42,960 --> 00:09:46,240
and reshuffles them.

119
00:09:46,240 --> 00:09:50,040
So if you have a state--

120
00:09:50,040 --> 00:09:52,290
so this is a simple
thing because then

121
00:09:52,290 --> 00:10:02,090
if you have a p alpha, this
alpha means really p alpha1,

122
00:10:02,090 --> 00:10:05,700
alpha2, all these things.

123
00:10:05,700 --> 00:10:08,340
Be alpha1, alpha2,
alpha3, alpha4.

124
00:10:08,340 --> 00:10:26,760
If those act on u1 1, u2
2, up to u n, capital n, n,

125
00:10:26,760 --> 00:10:28,860
what does it give you?

126
00:10:28,860 --> 00:10:32,490
Well, it says here
that you should

127
00:10:32,490 --> 00:10:41,290
put the 1 that has alpha 1
here on the first position.

128
00:10:41,290 --> 00:10:46,350
So you should put,
first, the u alpha1.

129
00:10:52,390 --> 00:10:57,730
And then you should
put next the u alpha2.

130
00:11:05,140 --> 00:11:10,950
And all the way up
to the u alpha n.

131
00:11:10,950 --> 00:11:15,550
That's in formulas,
the rule that I

132
00:11:15,550 --> 00:11:19,480
was giving you above there.

133
00:11:19,480 --> 00:11:25,270
It tells you how to move the
state from one to the other.

134
00:11:39,450 --> 00:11:44,700
You see, if alpha1
was 3, it would

135
00:11:44,700 --> 00:11:49,130
correspond to take the state
u3 and putting it first.

136
00:11:49,130 --> 00:11:54,080
So it's like that.

137
00:11:54,080 --> 00:11:57,080
So whatever alpha 1,
is whatever number,

138
00:11:57,080 --> 00:12:06,750
that state must be pulled
into the first [INAUDIBLE]..

139
00:12:06,750 --> 00:12:14,850
OK, so let's just think
a little about an example

140
00:12:14,850 --> 00:12:22,270
because it starts
getting interesting.

141
00:12:22,270 --> 00:12:27,970
So let's do three particles
for a little while, just

142
00:12:27,970 --> 00:12:29,200
to see what it is.

143
00:12:43,100 --> 00:12:49,020
So three particles, n equal 3.

144
00:12:49,020 --> 00:12:51,450
So what are the
permutation operators?

145
00:12:51,450 --> 00:12:55,955
You would say p123 is what?

146
00:12:58,500 --> 00:13:02,100
Put the first particle first,
the second particle second,

147
00:13:02,100 --> 00:13:03,600
the third particle third.

148
00:13:03,600 --> 00:13:05,010
Doesn't do anything.

149
00:13:05,010 --> 00:13:07,200
That is the identity.

150
00:13:07,200 --> 00:13:09,585
So that's the identity element.

151
00:13:12,810 --> 00:13:22,440
But then you could
cycle them, p321, p231.

152
00:13:22,440 --> 00:13:26,310
These are two more elements.

153
00:13:26,310 --> 00:13:32,190
This should have six elements
because it's 3 factorial.

154
00:13:32,190 --> 00:13:41,580
And now you have the following
one, p132, for example.

155
00:13:41,580 --> 00:13:43,620
That was not in the list.

156
00:13:43,620 --> 00:13:45,870
You had the p123.

157
00:13:45,870 --> 00:13:50,010
Now you flip this to p132.

158
00:13:50,010 --> 00:14:02,510
If I cycle them, p321,
and p213, these are it.

159
00:14:02,510 --> 00:14:07,340
The middle one, 312, yes.

160
00:14:07,340 --> 00:14:16,100
I was supposed to be cycling
them, 312 and 231, yes.

161
00:14:16,100 --> 00:14:18,280
Otherwise I would
have repeated that.

162
00:14:18,280 --> 00:14:21,020
Good eye.

163
00:14:21,020 --> 00:14:25,490
OK, so these are our operators.

164
00:14:25,490 --> 00:14:30,350
And these ones
are kind of funny.

165
00:14:30,350 --> 00:14:32,180
They are simpler ones.

166
00:14:32,180 --> 00:14:34,940
Please look at them.

167
00:14:34,940 --> 00:14:39,360
This operator says put the first
particle in the first position.

168
00:14:39,360 --> 00:14:43,265
So the first particle
is left unchanged.

169
00:14:43,265 --> 00:14:46,010
But the 3 and 2 are permuted.

170
00:14:46,010 --> 00:14:48,470
So people sometimes write this.

171
00:14:48,470 --> 00:14:51,980
They say, this permutation
operator is just

172
00:14:51,980 --> 00:14:57,560
a single transposition,
32 or 23.

173
00:14:57,560 --> 00:15:01,520
These two are transposed and
the first is left invariant.

174
00:15:01,520 --> 00:15:06,560
So this can be written as
a single transposition.

175
00:15:06,560 --> 00:15:10,910
In this permutation operator,
a 2 is left invariant.

176
00:15:10,910 --> 00:15:14,960
The second state is copied
to the second position.

177
00:15:14,960 --> 00:15:17,570
But the 1 and 3 are
exchanged, so this

178
00:15:17,570 --> 00:15:21,090
is a transposition of 1 and 3.

179
00:15:24,370 --> 00:15:28,170
And in this last
permutation operator,

180
00:15:28,170 --> 00:15:30,540
the third state
is left invariant

181
00:15:30,540 --> 00:15:31,960
on the third position.

182
00:15:31,960 --> 00:15:34,260
Nevertheless, the
second is put in first

183
00:15:34,260 --> 00:15:36,240
and the first is put in second.

184
00:15:36,240 --> 00:15:38,760
So it's a 12 transposition.

185
00:15:45,650 --> 00:15:52,970
OK, so the group has broken into
things that are transpositions

186
00:15:52,970 --> 00:15:56,990
and slightly more
complicated things.

187
00:15:56,990 --> 00:15:59,090
Now, let's look at
the transpositions.

188
00:16:01,820 --> 00:16:03,290
So these are transpositions.

189
00:16:14,070 --> 00:16:20,780
OK, what about the
transpositions?

190
00:16:20,780 --> 00:16:24,200
Transpositions are like
the permutation operators

191
00:16:24,200 --> 00:16:27,110
we were doing before
of two particles.

192
00:16:27,110 --> 00:16:29,860
You're just changing
two particles.

193
00:16:29,860 --> 00:16:34,060
You're leaving the
other ones unchanged.

194
00:16:34,060 --> 00:16:40,850
So these transpositions
are Hermitian and unitary.

195
00:16:40,850 --> 00:16:55,710
Positions are
Hermitian and unitary,

196
00:16:55,710 --> 00:17:01,500
just like we had p21 before.

197
00:17:09,750 --> 00:17:11,890
Now here comes the
first statement

198
00:17:11,890 --> 00:17:17,260
that is quite remarkable
about this permutation group.

199
00:17:17,260 --> 00:17:22,150
Any permutation, now we
claim, is the product

200
00:17:22,150 --> 00:17:25,089
of some transpositions.

201
00:17:25,089 --> 00:17:29,980
You see, if somebody gives you
a permutation of n integers,

202
00:17:29,980 --> 00:17:33,700
I think you're convinced that
by changing two at a time,

203
00:17:33,700 --> 00:17:36,460
you can reach that.

204
00:17:36,460 --> 00:17:40,840
So that means that with
transpositions, you can

205
00:17:40,840 --> 00:17:43,060
reach every possible element.

206
00:17:43,060 --> 00:17:51,370
So any-- this is very
important-- any permutation is

207
00:17:51,370 --> 00:17:57,540
the product of transpositions.

208
00:18:04,050 --> 00:18:10,350
And that implies, in a
sense, the main result

209
00:18:10,350 --> 00:18:14,220
that you want from operators
in this Hilbert space.

210
00:18:14,220 --> 00:18:20,220
Since transpositions are
unitary and any permutation

211
00:18:20,220 --> 00:18:26,400
is a product of transpositions,
any permutation is unitary.

212
00:18:26,400 --> 00:18:35,530
So all permutations are unitary.

213
00:18:41,950 --> 00:18:48,370
It's not true that all
permutations are Hermitian.

214
00:18:48,370 --> 00:18:49,960
You see, when you
have the product

215
00:18:49,960 --> 00:18:52,870
of her Hermitian
operators the product

216
00:18:52,870 --> 00:18:57,670
of her Hermitian operators
is not necessarily Hermitian.

217
00:18:57,670 --> 00:18:59,096
You've seen that.

218
00:18:59,096 --> 00:19:00,970
If you have two operators
that are Hermitian,

219
00:19:00,970 --> 00:19:02,890
the order changes.

220
00:19:02,890 --> 00:19:09,700
And the product is not
Hermitian, x and p.

221
00:19:09,700 --> 00:19:13,070
Hermitian operators, the
product is not Hermitian.

222
00:19:13,070 --> 00:19:19,690
So all permutations are
unitary but are not Hermitian.

223
00:19:19,690 --> 00:19:23,700
And the third concept
that comes here

224
00:19:23,700 --> 00:19:29,440
illustrated [INAUDIBLE] also
a very fundamental concept.

225
00:19:29,440 --> 00:19:34,940
We said that any permutation
is a product of transpositions.

226
00:19:37,890 --> 00:19:41,400
But it's not a unique
product of transposition.

227
00:19:41,400 --> 00:19:43,050
In fact, you could--

228
00:19:43,050 --> 00:19:46,920
if you are given
some order of objects

229
00:19:46,920 --> 00:19:50,190
and you're asked to rearrange
them by transposition,

230
00:19:50,190 --> 00:19:53,130
you may do them in
different order.

231
00:19:53,130 --> 00:19:57,000
So the product is not
uniquely determined.

232
00:20:00,120 --> 00:20:03,600
The product is,
in fact, you could

233
00:20:03,600 --> 00:20:08,640
find somebody that does a
permutation with some number

234
00:20:08,640 --> 00:20:11,940
of transpositions
and another person

235
00:20:11,940 --> 00:20:14,250
with another set
of transpositions.

236
00:20:14,250 --> 00:20:17,790
But one thing that
has to happen--

237
00:20:17,790 --> 00:20:20,760
and it's always the
same-- is that the number

238
00:20:20,760 --> 00:20:26,640
of transpositions is the
same [INAUDIBLE] two.

239
00:20:26,640 --> 00:20:31,080
That is, if you find that to
get a permutation you need

240
00:20:31,080 --> 00:20:33,690
an even number of
transpositions,

241
00:20:33,690 --> 00:20:41,460
everybody else will need also an
even number of transpositions.

242
00:20:41,460 --> 00:20:44,280
If you find an odd
number of transpositions,

243
00:20:44,280 --> 00:20:48,030
everybody will find an odd
number of transpositions.

244
00:20:48,030 --> 00:20:51,240
And that's kind of
clear if you have--

245
00:20:51,240 --> 00:20:55,950
if you've achieved
the desired order

246
00:20:55,950 --> 00:20:59,040
by doing some number
of transpositions,

247
00:20:59,040 --> 00:21:01,980
you can add two
more transpositions

248
00:21:01,980 --> 00:21:03,330
and not change anything.

249
00:21:03,330 --> 00:21:06,600
You can flip two objects
and flip them back.

250
00:21:06,600 --> 00:21:11,040
So this is a pretty
important result as well.

251
00:21:11,040 --> 00:21:17,280
So it means that permutations
are either even or are odd.

252
00:21:17,280 --> 00:21:35,990
So permutations
are even, if built

253
00:21:35,990 --> 00:21:48,740
with an even number
of transpositions

254
00:21:48,740 --> 00:22:01,880
or odd if built with an odd
number of transpositions.

255
00:22:01,880 --> 00:22:07,160
So two kinds of permutations,
even permutations

256
00:22:07,160 --> 00:22:08,630
or odd permutations.

257
00:22:13,010 --> 00:22:16,760
So all these things
we've kind of

258
00:22:16,760 --> 00:22:21,050
seen with this
permutation group.

259
00:22:21,050 --> 00:22:28,040
There's one more thing
that is necessary to note.

260
00:22:31,170 --> 00:22:36,750
Something that may have
seemed that coincidence there.

261
00:22:36,750 --> 00:22:44,170
In fact, the 3 on the
right are transpositions.

262
00:22:44,170 --> 00:22:46,320
And they're a single
transposition each.

263
00:22:46,320 --> 00:22:52,600
So each one of those on the
right is an odd permutation.

264
00:22:52,600 --> 00:22:55,000
It's built with
one transposition.

265
00:22:55,000 --> 00:22:59,980
On the other hand, the other
ones are even transpositions,

266
00:22:59,980 --> 00:23:02,500
the first three.

267
00:23:02,500 --> 00:23:06,070
The first one is built
with zero transpositions

268
00:23:06,070 --> 00:23:10,930
but the others are built
with two transpositions.

269
00:23:10,930 --> 00:23:14,740
It's something you could
just see in detail.

270
00:23:14,740 --> 00:23:18,310
In fact, I would
say it's worth doing

271
00:23:18,310 --> 00:23:23,140
the table of multiplication
of this group.

272
00:23:23,140 --> 00:23:30,370
But that coincidence is,
in fact, true in any--

273
00:23:30,370 --> 00:23:51,360
in sn, in the group sn, the
number of even permutations

274
00:23:51,360 --> 00:23:59,860
is equal to the number
of odd permutations.

275
00:24:06,180 --> 00:24:10,680
Some pretty nice result.
There's equal numbers of even

276
00:24:10,680 --> 00:24:16,620
and odd permutations in
any permutation group.

277
00:24:16,620 --> 00:24:20,970
Actually, it's surprisingly
simple to show that.

278
00:24:20,970 --> 00:24:24,210
You think of the even
permutations here-- here are

279
00:24:24,210 --> 00:24:26,760
the even permutations.

280
00:24:26,760 --> 00:24:28,800
And here are the
odd permutations.

281
00:24:34,070 --> 00:24:38,010
And you can easily
show that there

282
00:24:38,010 --> 00:24:40,640
is equal number on this one.

283
00:24:40,640 --> 00:24:43,020
So let do it.

284
00:24:43,020 --> 00:24:49,190
One way is to take the
permutation element.

285
00:24:49,190 --> 00:24:53,640
So suppose you have
a permutation--

286
00:24:53,640 --> 00:24:55,940
how should I call it--

287
00:24:55,940 --> 00:24:58,640
p21.

288
00:24:58,640 --> 00:25:01,620
You leave everything the
same but you have p21.

289
00:25:01,620 --> 00:25:07,650
That permutes the
second and first labels.

290
00:25:07,650 --> 00:25:10,880
Since it's an n group, the
rest are left invariant,

291
00:25:10,880 --> 00:25:13,910
so 3, 4, 5, 6, 7, 8, 9, 10.

292
00:25:13,910 --> 00:25:21,740
And you say, this maps
any permutation here

293
00:25:21,740 --> 00:25:23,540
into a permutation there.

294
00:25:23,540 --> 00:25:24,440
Why?

295
00:25:24,440 --> 00:25:29,360
Because if you have an
even permutation here,

296
00:25:29,360 --> 00:25:35,570
any element here has an even
number of transient mutations,

297
00:25:35,570 --> 00:25:37,470
of transpositions.

298
00:25:37,470 --> 00:25:40,300
So this is another
transposition,

299
00:25:40,300 --> 00:25:42,110
it transposes 2 and 1.

300
00:25:42,110 --> 00:25:47,480
So when you multiply
an element here by p21,

301
00:25:47,480 --> 00:25:49,660
if it has an even number
of transpositions,

302
00:25:49,660 --> 00:25:51,050
now it has an odd one.

303
00:25:51,050 --> 00:25:55,910
So it must land somewhere here.

304
00:25:55,910 --> 00:26:03,995
So any-- this p21
maps the even--

305
00:26:06,540 --> 00:26:12,510
the even permutations
to the odd permutations.

306
00:26:18,460 --> 00:26:25,750
Moreover, it maps them in
what's called 1 to 1 fashion.

307
00:26:25,750 --> 00:26:29,320
If you have two permutations
here that are different,

308
00:26:29,320 --> 00:26:33,220
it will map to two permutations
here that are different.

309
00:26:33,220 --> 00:26:33,740
Why?

310
00:26:33,740 --> 00:26:41,120
Because suppose you have here
pi and pj, two permutations.

311
00:26:41,120 --> 00:26:50,260
1 is mapped to p21 pi, and
the other is map to p21 pj.

312
00:26:50,260 --> 00:26:56,380
But if pi and pj
are different, two

313
00:26:56,380 --> 00:27:01,180
permutations here,
I and j, pi and pj.

314
00:27:01,180 --> 00:27:05,020
And if they are
different, then it

315
00:27:05,020 --> 00:27:09,070
follows that these two are
different as well after you

316
00:27:09,070 --> 00:27:11,590
multiply them by p21.

317
00:27:11,590 --> 00:27:12,730
Why?

318
00:27:12,730 --> 00:27:17,380
Because if they were
equal, p21 has an inverse.

319
00:27:17,380 --> 00:27:21,100
It is itself p21, so you
could multiply them in.

320
00:27:21,100 --> 00:27:25,030
And then you would show
that pi is equal to pj.

321
00:27:25,030 --> 00:27:27,250
And that can't happen.

322
00:27:27,250 --> 00:27:31,090
So two different
elements here will

323
00:27:31,090 --> 00:27:34,490
go to two different
elements there.

324
00:27:34,490 --> 00:27:39,460
So if you think of all
your elements here, go--

325
00:27:39,460 --> 00:27:44,110
if you have 20 elements here,
they go to 20 elements here.

326
00:27:44,110 --> 00:27:47,530
The only thing that can happen
is that some elements are not

327
00:27:47,530 --> 00:27:49,750
reached here.

328
00:27:49,750 --> 00:27:52,880
That would mean that these
two sets are not the same.

329
00:27:52,880 --> 00:27:57,250
So you need to show that
this map reaches everybody.

330
00:27:57,250 --> 00:28:02,665
So this shows it's
a one to one map.

331
00:28:07,540 --> 00:28:10,210
And in fact, it's
kind of a joke,

332
00:28:10,210 --> 00:28:15,100
but in a way to think
of this, one to one

333
00:28:15,100 --> 00:28:18,490
really means two to
two, in the sense

334
00:28:18,490 --> 00:28:22,040
that two things
go to two things.

335
00:28:22,040 --> 00:28:25,660
So it's kind of a
funny name, one to one.

336
00:28:25,660 --> 00:28:27,370
So the other thing
is that you need

337
00:28:27,370 --> 00:28:29,590
to show that everybody reaches.

338
00:28:29,590 --> 00:28:31,250
So the map is--

339
00:28:31,250 --> 00:28:35,260
let's call surjective,
everybody reaches.

340
00:28:35,260 --> 00:28:38,110
So it's clear that
everybody reaches

341
00:28:38,110 --> 00:28:47,620
because if you have a pk
in here, permutation pk,

342
00:28:47,620 --> 00:28:54,960
this pk can be written as
p21 times p21 times pk.

343
00:28:54,960 --> 00:28:57,640
Because p21 times p21 is 1.

344
00:29:00,200 --> 00:29:08,000
And this pk, since this
was an odd permutation,

345
00:29:08,000 --> 00:29:11,610
this is an even one.

346
00:29:11,610 --> 00:29:14,370
And then you've shown
that pk is obtained

347
00:29:14,370 --> 00:29:17,790
by acting with p21 on
some even permutation.

348
00:29:17,790 --> 00:29:19,320
So you'll reach.

349
00:29:19,320 --> 00:29:21,826
So the map is surjective.

350
00:29:26,050 --> 00:29:29,500
And therefore these two
things are the same.

351
00:29:29,500 --> 00:29:34,270
They're identical numbers of
even and odd permutations,

352
00:29:34,270 --> 00:29:38,844
a very nice fact
about these groups.