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Keywords:
Hopf group-coalgebra; Radford's $\pi $-biproduct; automorphism
Hopf group-coalgebra; Radford's $\pi $-biproduct; automorphism
Summary:
We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford's $\pi $-biproduct. Firstly, we discuss the endomorphism monoid ${\rm End}_{\pi \text {-Hopf}}(A\times \nobreak H, p)$ and the automorphism group ${\rm Aut}_{\pi \text {-Hopf}}(A\times H, p)$ of Radford's $\pi $-biproduct $A \times H =\{A \times H_\alpha \}_{\alpha \in \pi }$, and prove that the automorphism has a factorization closely related to the factors $A$ and $H=\{H_\alpha \}_{\alpha \in \pi }$. What's more interesting is that a pair of maps $(F_L,F_R)$ can be used to describe a family of mappings $F=\{F_\alpha \}_{\alpha \in \pi }$. Secondly, we consider the relationship between the automorphism group ${\rm Aut}_{\pi \text {-Hopf}}(A\times H, p)$ and the automorphism group ${\rm Aut}_{\pi \text {-}\mathcal {Y}\mathcal {D}\text {-Hopf}}(A)$ of $A$, and a normal subgroup of the automorphism group ${\rm Aut}_{\pi \text {-Hopf}}(A\times H, p)$. Finally, we specifically describe the automorphism group of an example.
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