1

Compute the order and inverse of $(12)(43)(13542)(15)(13)(23)$.

2

How many permutations of order 2 are there in $S_{n}$?

3

$GL(2,Q)$ denotes the set of the nonsingular (having nonzero determinant, 行列式非 0) rational square matrix of length 2 under matrix product.

1. Show that it’s a group.
2. Compute the order of A=[0, -1; 1, 0] and B=[0, 1; -1, 1] .
3. What’s the order of AB ?

(1)

$\forall A \in GL(2,Q),\mid A \mid \neq 0$，且A为方阵，因此存在$A^{-1}$，使得$A^{-1}A=E$

(2)
A的阶为4
B的阶为6
(3)

4

Under matrix product, the special linear group is defined as:

Prove that it’s a subgroup of $GL(2,R)$.

5

1. The stochastic group $\sum (n,R)$ is a subset of $GL(n,R)$ that each column sums to 1. Prove that it’s a group.
2. The doubly stochastic group $\sum ‘(n,R)$ is a subset of $GL(n,R)$ that each column and each row sum to 1. Prove that it’s a subgroup of $\sum (n,R)$.

(1)

(2)

7

Show that determinant is a surjective (but not injective) homomorphism, from matrix product on non-singular $k \times k$ matrices to multiplication on non-zero real numbers

Note: $\mathbf{ker} det$ is the special linear group $SL(k, \mathbb{R} ) = \lbrace A \in GL(k, \mathbb{R} ):det(A)=1 \rbrace$.

8

The real addition group $(\mathbb{R},+)$ is mapped to $H=((0,1),*)$ on interval $(0,1)$ by sigmoid function $f(x)= \frac{1}{1+e^{-x}}$, where the operation $\ast$ is:

1. Does H have the 1-element? What is it?
2. Does every element in H have an inverse element?
3. Is the sigmoid function between G and H an isomorphism?

(1)

(2)

(3)

9

Group $( \mathbb{R} ,+)$ is mapped to $K=(\mathbb{R}^{+}, \&)$ by softplus function $f(x)=ln(1+e^{x})$.

1. Find a proper operation & which makes it a homomorphism.
2. Prove that K is an abelian group with the above &.

(1)

(2)

10

Show that for any $n \in \mathbb{Z}$, $(n \mathbb{Z},+) \vartriangleleft (\mathbb{Z},+)$, and $\mathbb{Z} /n \mathbb{Z}$ is $\mathbb{I}_{n}$

11

If $N \vartriangleleft G$, then for any subgroup H of group G, HN is also a subgroup.

12

If H and K are subgroups of a group G, then

1. HK is a subgroup iff HK=KH
2. If G is an abelian group, and $H \cap K= \lbrace 1 \rbrace$ , then $HK \cong H \times K$

(1)

(2)