hi a little help would be kindly appreciated here guys.
any suggestions on how to go about doing these?
INFORMATION
-----------------------
if K,Q are groups is a homomorphism the semi direct product is defined as follows.
(i) as a set
(ii) the group operation * is
THE QUESTION
-----------------------
Verify formally that is a group and find a formula for in terms of and
-----> to show that it is a group, i know i have to show that the 4 conditions for being a group (e.g. associativity, closure, existance of identity element, existance of inverse) have to be satisfied. but not really too sure how to show it.. and im completely baffled for the 2nd part of the question.
please help out thnx
thnx a lot for the help..
here is my working out so far: please verify and suggest any corrections for wrong the parts.
Checking of the 4 conditions for a group
--------------------------------------
Closure:
.
Since is an automorphism of K, for some . Since K is a group, . Similarly, . So and closure holds.
Existance of identity element:
and
.
So .
So the identity element exists.
Existance of inverse element:
The inverse is an element such that and
.
so
So the inverse element exists.
Associativity:
the 2 answers dont seem to match up here. please verify this.
2ND PART OF QUESTION
Finding the inverse for .
The inverse is an element such that . So . This means . And