# 代數中群論的兩個問題

Q : Find all groups of order 77 up to isomorphism?

Q : List all abelian groups of order 360?

Rating

1 For |G| = pq with p < q and q not equal to 1mod p, G is cyclic of order pq. Hence, in particular, there is only one isomorphism class of groups of such orders pq.

Since 77=7*11 and 11 not equal to 1mod 7,there is only one isomorphism class of groups of such orders 77

2

For every finite abelian n=p1^r1 p2^r2 pn^rn

can be written as

Zn~Zp1^r1 Zp2^r2 Zpn^rn ,

where the p’s are primes and the r’s are positive integers (primary decomposition).

Alternatively, you can write the same group as

Zd1 Zd2 Zdm,

where the d’s are positive integers and d1 | | dm (invariant factor decomposition).

Now 360=2^3*3^2*5

So, the all abelian groups of order 360 are

primary decomposition

Z2 Z2 Z2 Z3 Z3 Z5

Z2 Z2 Z2 Z9 Z5

Z2 Z4 Z3 Z3 Z5

Z2 Z4 Z9 Z5

Z8 Z3 Z3 Z5

Z8 Z9 Z5

invariant factor decomposition

Z2 Z6 Z30

Z2 Z2 Z90

Z6 Z60

Z2 Z180

Z3 Z120

Z360

• 蜉蝣
Lv 6

應該沒這麼簡單

是用sylow定理沒錯

• 費瑪
Lv 4

Q1：Z_77

Q2：

Z_8*Z_9*Z_5

Z_2*Z_4*Z_9*Z_5

Z_2*Z_2*Z_2*Z_9*Z_5

Z_8*Z_3*Z_3*Z_5

Z_2*Z_4*Z_3*Z_3*Z_5

Z_2*Z_2*Z_2*Z_3*Z_3*Z_5