Isomorphism of Groups and Automorphisms

This text discusses the isomorphism of groups and automorphisms, with a focus on examples and proofs. It covers topics such as the order of elements in groups, the number of automorphisms in Z12, error correction in codes, and the classification of actions of discrete abelian or finite groups on AFD factors. Additionally, it explores the concept of centrally trivial automorphisms and their relation to pointwise inner automorphisms.

Ans 1.
Solution (a) Every element of Z2 x Z2 x Z2 has order 1 or 2. For example, ([1], [0], [1])2 ([1] e [1] , [0] e
[0], [1] e [1] ) Go] , [0], [0] ) is the identity element of Z2 x 7Z2 x Z2, SO ([1], [0], [1]) has order 2. There's
an element of Z2 x Z4 of order 4, namely ([0]2, [1]4), but there's no element of order 8 in Z2 X Z4. The
element [1]8 of Z8 has order 8. Hence the three groups 7G2 x 7G2 x Z2, 7G2 x Z4 and Z8 are not
isomorphic, by Theorem 41(d). On the other hand, it's easy to show that G x H is always isomorphic to H
x G, since the mapping (g, h) (h, g) is an isomorphism. [Check this!] So Z2 x Z4 c 7Z4 x Z2. And Z4 x Z2 is
not isomorphic to any other group in the list, since if it were then Z2 x Z4 would be too (by transitivity of
Ras), but we've already shown that this is not true.
Ans 2.
Consider Z12. We have 4)(n) = 4 since 1, 5, 7 and 11 are the only integers r such that 1 r < 12 and (r, 12) =
1. Therefore, there are exactly four auto-morphisms of Z12 and are given below. = I, the identity map of
76122 rs : Z12 -' Z12 defined by f5(m) = 5m (mod 12), ri : Z12 -> Z12 defined by /7(m) = 7m (mod 12), fii :
Z12 -> 712 defined by fit(m) = 1 lm (mod 12).
The following table gives a complete description of all the four automor-phisms of 7612.
0 1 2 3 4 5 6 7 8 9 10 11
F1 0 1 2 3 4 5 6 7 8 9 10 11
F2 0 5 10 3 8 1 6 11 4 9 2 7
F3 0 7 2 9 4 11 6 1 8 3 10 5
F4 0 11 10 9 8 7 6 5 4 3 2 1
Ans 3. 1. dmin(c)=5
2. C can detect 3 error at max
3. C can correct 2 error at max
4. Decoded word is 1111011001011
Ans 4. S2(c): {1010,0010,1110,1000,1011,1001,1111,0101,0110,0011,1100}
Ans 5 . S3(c): ∑
k=0
12
Ck
n
2. If c can correct 3 error so maximum number of codeword in c is 6.
Ans 6. is a finite, non-empty set of states;
 is a finite, non-empty set of tape alphabet symbols;
 is the blank symbol (the only symbol allowed to occur on the tape infinitely often at any step
during the computation);

 is the set of input symbols, that is, the set of symbols allowed to appear in the initial tape
contents;
 is the initial state;
An automaton is the one that represented by (Q, ∑, δ, q0, F), where −
 finite set of states .
 finite set of symbols, alphabet of the automaton.
 transition function.
 initial state where any input is processed.
 set of final state.
Ans 8. An automorphism a of ~ is centrally trivial if and only if a is of the form
a=Ad(u).O~, where 0~, is an extended modular automorphism for a dominant weight cp
on d~, c is a O-cocycle on ~ and u E ~
We give a complete proof of this characterization in w 3. The centrally trivial
automorphisms are also related to pointwise inner automorphism of Haagerup-StCrmer
[14].
In the classification of discrete amenable group actions on AFD factors of type III
in [20], the case of type 1111 was left open. Here we now classify actions of discrete
abelian groups and finite groups on the AFD factor of type 1111. Thus the classification
of actions of discrete abelian or finite groups is complete, and this will be enough to
accomplish classification of compact abelian group actions on AFD factors in Kawahigashi-
Takesaki [17] along the lines of Jones-Takesaki [16] and Suthedand-Takesaki
[20].