TITLE:
Study on the Automorphism Group Structure of Several Typical Groups
AUTHORS:
Renbing Xiao
KEYWORDS:
Automorphism Group, Symmetric Group, Dihedral Group, Quaternion Group
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.15 No.12,
December
23,
2025
ABSTRACT: This paper provides a systematic and unified expository study on the automorphism group structures of three fundamental finite groups in modern algebra: the symmetric group on three elements
S
3
, the dihedral group of order
D
8
, and the quaternion group
Q
8
. By selecting characteristic subgroups of the groups, constructing a homomorphism from the automorphism group to the permutation group of the set of characteristic subgroups, and combining methods such as kernel analysis and order matching, we rigorously prove three well-established isomorphisms:
Aut(
S
3
)≅
S
3
,
Aut(
D
8
)≅
D
8
,
Aut(
Q
8
)≅
S
4
. The proof process adheres to a consistent logical framework with complete details, serving as a pedagogically valuable resource for understanding the construction of automorphism groups of finite groups. While the results themselves are foundational in group theory, this paper unifies scattered proof methods from introductory abstract algebra materials, facilitating intuitive comparison and learning for students and educators.